AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS, VOLUME VII ALGEBRAIC ARITHMETIC BY ERIC T¢BELL PROFESSOR OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY NEW YORK PUBLISHED BY THE AMERICAN MATHEMATICAL SOCIETY 501 WEST 116TH STREET 1927 GERMANY ’ MBURG rr, Ha = E 2 = 2 iS B= BD — CONTENTS INTRODUCHON. sa ss aa a 1 CHAPTER 1 VARIETIES OF ALGEBRA USEFUL IN ALGEBRAIC ARITHMETIC 1- 3. Irregular fields, modules, rays, rings, semigroups 5 4— 7. Characteristics of algebraic arithmetic ........ 9 8-9. One-rowed matrices. ....... ou 20 0... 0, 15 10-12. Maric fields... .. 0. 0 i een. on 17 13-18. The associated functional varieties of U,, .. ... 20 19-23. The irregular fields B, €, D associated with A. 27 24-26. The partitions of a matrix... ................ 30 CHAPTER 11 THE ALGEBRA Lf OF PARITY 1-6. Absolute and relative parity. ........... 0... 34 1-9. Abstract identity of 5 with the algebra T of the circular functions... 0... ii iri ah 43 10-14. Expansion and decomposition in B........... 47 15-17. The identical transformations in B........... 51 18-21. Divisibility in. B...... inl. i b4 22, . Algebraic parity, generalization of B......... 63 CHAPTER 111 THE ALGEBRAIC ARITHMETIC OF MULTIPLY PERIODIC FUNCTIONS Y- 6. The prineiple of paraphvase .........-.. 0... ., 64 6.1-11. Extension of the principle to higher forms .... 80 12-16. Application of the principle to theta quotients. 88 17-20. Application to theta functions of » >>1 arguments 106 111 €44471 1-9. 10-12. 13-14. 15-17%, CONTENTS. CHAPTER IV APPLICATIONS OF THE ALGEBRAS €, © PAGE Algebra 8, ee a 112 Phe variety €. of €.- =... 000. 124 Extensions and further instances of € ........ 144 Applications of € to the algebra of sequences . 146 CHAPTER V ARITHMETICAL STRUCTURE Nature of general arithmetic... ....... ....... 160 Meithmetic- 2v of Lo 0 v0 dan 165 Avithmetization .. ....... ui. oa 175 ALGEBRAIC ARITHMETIC INTRODUCTION 1. Intermediate between the modern analytic theory of numbers and classic arithmetic as developed by the school of Gauss, is an extensive region of the theory of numbers where the methods of algebra and analysis are freely used to yield relations between integers expressed wholly in finite terms and without reference, in the final propositions, to the operations or concepts of limiting processes. This part of the theory of numbers we shall call algebraic arithmetic. Its boundaries are not sharply defined, nor is it desirable that they should be, as power comes from flexibility and almost any part of arithmetic can be profitably employed in any other. Nevertheless there is a large, growing and somewhat uncoordinated body of results, with many aims and methods peculiar to itself, which falls into neither the classic theory of numbers nor the modern developments of the analytic and algebraic theories, and this region, which we have called algebraic arithmetic, offers many suggestive opportunities for systematic exploration. It is the purpose of the following chapters to outline a few promising directions in which progress may be made toward classifying, extending and generalizing the methods and results of algebraic arithmetic. The in- sistence will be upon general methods rather than specific applications, as the latter are so numerous, and so readily made from the general formulations, that it will be sufficient merely to indicate occasionally a few of them to lend con- creteness to the abstract theories. What is given here is but a narrow cross section of a very extensive field. 1 1 2 ALGEBRAIC ARITHMETIC. 2. The distinction between analytic and algebraic arithmetic is evident from the comparison of two famous theorems. Let F(n) = the number of representations of » as a sum of 4 integer squares, s(n) = the sum of all the odd divisors of n; n(x) = the number of primes < z. Then : 1 Fn), fe BIZ + fr 1)7] s(n), zr TT (z)/(x/1log x) = 5 ie fist accords’ ith the Sonal description in § 1; the © “second belongs to a totally different order of ideas. They have in common one feature however which is of more than merely historical interest: both were discovered by tran- scendental methods. The first has since been proved in many ways in the manner traditionally called elementary in arith- metic; from its nature the second is incapable of such proof. The numerous proofs which have been devised for the 4-square theorem, or for its less complete form which asserts that every integer is the sum of 4 squares, certainly one of the perfect gems of arithmetic, emphasize the advantages in the theory of numbers of multiplying proofs for the light which such revaluations of known theorems throw upon the theories in which they originate. This particular theorem has enriched, and has been enriched by the theories of elliptic functions, quaternions and their generalizations, and Dickson arithmetics. Starting from Jacobi’s elementary re- casting of the transcendental proof by means of elliptic functions, for instance, Liouville states that, following Dirichlet, he was led to the discovery of his powerful general formulas in the theory of numbers which he published without proofs, and thence back, by a natural reaction, to an interesting generalization of elliptic functions — which has not yet been fully explored. Again, Euler's early attempts to prove that every integer is the sum of 4 squares gave him his identity which, in modern phrase, is the theorem for the norm of the product of two quaternions; this in turn suggested to Graves, Roberts, Cayley and others purely algebraic investigations which have again reacted on the theory of numbers through Dickson's arithmetics. INTRODUCTION. 3 3. From our point of view in the present theory it is little to the purpose to offer as an objection against a transcendental proof the remark that a purely elementary demonstration is in existence or may be found. A transcendental proof often exhibits the advantages inherent in the impulses to general- ization and unification characteristic of analysis. But it must not be forgotten, on the contrary, that a finite, strictly ele- mentary proof as frequently fertilizes an otherwise barren waste of algebra by the introduction of new arithmetical concepts or processes. Neither the transcendental nor the elementary method can be fruitfully used for long to the exclusion of the other. If occasionally we seem to go out of our way to make the elementary transcendental our apology is that analytical proofs, judged by the ease with which they are retained and applied, are altogether more elementary than many in the theory of numbers which depend upon nothing more advanced than the rudiments of algebra. The principal reason however for our emphasis of transcendental methods is their power, suggestiveness, and the readiness with -which they yield themselves to generalization. Incident- ally it may be remarked that even the most elementary proofs in the theory of numbers are tainted by the transcendence inherent in mathematical induction and in the notion of “any integer nn’. 4. In the description of algebraic arithmetic we have used several terms whose significance is usually taken by consent as being obvious, but which will be clearly understood only when large tracts of extant arithmetic have been subjected to the postulational method. Among these is arithmetic itself. It is generally conceded that an arithmetic, as distinguished from an algebra, must be based on integral elements. There exists however no set of postulates sufficiently elastic to embrace all the known instances of elements called integral by their creators. An abstract logical analysis, culminating in the relational formulation of existing arithmetical theories should disclose the essential characteristics common to all. In the absence of even partial analysis of this kind for the 1* 4 ALGEBRAIC ARITHMETIC. fundamental concepts of arithmetic we shall offer tentatively a definition to justify our subsequent characterization of certain theories as arithmetical, and to this we turn first. The algebraic varieties (rings, irregular fields, matric fields, etc.,) encountered in this attempt are valuable instruments in algebraic arithmetic, independently of whether they may ultimately yield a satisfactory description of arithmetic itself. We shall therefore state their postulate systems in full. For easy reference the several varieties indicated by Ger- man capitals have been included in the index at the end of the book. CHAPTER 1 VARIETIES OF ALGEBRA USEFUL IN ALGEBRAIC ARITHMETIC IRREGULAR FIELDS, MODULES, RAYS, RINGS, SEMIGROUPS, §§ 1-3 1. Common algebra, abstract identity. By common algebra A we shall mean the set of all propositions, including 77, implied by the set 77 of postulates of the abstract field J in which the notations are as usual: a+b, ab denote the sum, product of any two elements a,b in §; the zero, unity in § are written 0,1, and XZ is as in Dickson, Algebras and their Arithmetics (1st edition, p.200). This #7 is identical with that stated in § 2 if there we take m =—1. The fields of all complex, rational numbers are denoted by F., J» respectively. The elements and operations of 2 are abstract in the sense of marks without significance beyond that implied by the assertion of 7. By assigning to the elements and operations of A specific interpretations 7; (j = 1, 2, -..) such that the resulting systems 2; are consistent with 77 (and with 7), we obtain instances A; of A, and A;, Ax are said to be identical or distinct according as Ij, I are the same or different; A and distinct A; (j =1, 2, ...) are said to be abstractly identical. The simplest 2; are F., T,. A system obtained from 2 by modification (including sup- pression) of one or more of the postulates 7 will be called a variety. In any variety 8 definitions precisely similar to those for A concerning instances and abstract identity are presupposed. Varieties will be designated by German capitals. 2. Irregular fields. 11 consists of a set = of elements «, B,---,7,--- and two operations §, P (called addition, multiplication) which may be performed upon any two distinct or identical elements «, 8 of =, in this order, to produce 5 6 ALGEBRAIC ARITHMETIC. uniquely determined elements S{e, 8}, Plea, 8} in = (this is a postulate), such that the postulates (2.1)-(2.5) are satisfied. Elements of = will be called elements of 1, and similarly in all like cases. (2.1) If «, 8 are any two elements of 1, then S{8, a} = Sle, 8}, P{8 oc} = Plo, 8}. (2.2) If a, 8, y are any three elements of 1, then S{8{e, 8,7} — Sle, 58,7}, P|Ple,ghi} = Pla. Pls}, Plea, S18, 7 m= §{Ple, At, Plo, 7H. (2.3) There exist in U two distinct elements, denoted by €, v, such that, if « is any element of 11, Ste, ct = q, Ple, v} m=) (2.4) Whatever be the element « of 1 there exists in U an element «' such that S{e, «'} = ¢. (2.5) Whatever be the element 8 of 11 different from each of the m (=1) distinct elements §; (j =0, ---, m—1), where lo=2C, of 1, there exists in U an element A such that Pls. BL =v, The § (j=0,..-,m—1) in (2.5) are called irregular, all other elements of WU regular; Sie, 8), Pla, 8} in (2.1) are called the sum, product of «, 8, and {, v in (2.3) the zero, unity of Ul. An instance of U is said to be regular or urregular according as m =1 or m >>1; when 1 contains precisely m irregular elements we indicate this by writing 1,,, and instances of Il are designated by accents, thus 11, U,, ---. Hence U; = A=F. We shall write U, = J, and call SE an regular field. By slight modifications of Dickson's proofs for § we have the following basic theorems for 11,, and hence for J. (2.6) £, v are unique in W,. 2.7) Sle, 8} = Sle, r} implies 8 = y. Hence « in (24) is unique; it is called the negative of «. VARIETIES OF ALGEBRA. 7 (2.8) If «, 8 are in WU, there exists in 11,, a unique § such that S{e, & = 8. CO Tie B 7 ave in WN, and Plea; 8 = Pla, 7}, and if « is regular, then 8 = y. Hence #8’ in (2.5) is unique. (2.10) If «, 8 are in U,, and « is regular, there exists in U,, a unique 0 such that Ple, 0} = #8; 0 is called the quotient of 8 by a; if 8 = v, 0 is called the reciprocal of e. It is sufficient, and frequently shorter, to replace division by « by multiplication by the reciprocal of «. To show that division in a given variety is possible in the sense of (2.10) it suffices to prove that a regular element has a unique reciprocal. This remark will be found useful in some of the more com- plicated instances occurring later. The consistency of the postulates is proved as we proceed by exhibiting numerous instances of varieties which satisfy them. For the following remarks on ,, which place 1, with respect to linear algebra, I am indebted to Professor Wedder- burn. Let & be a cummutative linear associative algebra, of which ‘the algebra. @ == (5, 5, 2+, bniy), With =, is an invariant subalgebra. Then 1,, is the difference algebra &—2, and equality in U,, can be interpreted as “congruent mod £27, also v is the identity element of &. If 2 is maximal and & commutative, U,, = ¥— 2 is necessarily a field. The varieties F, J& and several of their instances §, SF ++ - are fundamental for algebraic arithmetic, as also are the following. 3. Modules, rays, rings, commutative semigroups. Denote for the moment by S, P, P_;, S_; addition, multi- plication, division and subtraction in U,, Pi, S_; being given by (2.10), (2.7), and write (OT...) for a variety closed under operations 0, 7 ..., together with those operations. Then in U,, we have 15 conceivable subvarieties (S“ P’ P%; 8%), where each of a, b, ¢, d = 0 or 1. We shall require only the following in addition to U,,: the module, M = (S_1), — (8S_1); the ray, (PP_1); the ring, R = (SPS_;). Thus in A we have the varieties commonly designated by the same names; 8 ALGEBRAIC ARITHMETIC. a module of A is a set in A closed under addition and sub- traction in A; a ray of A is an abelian group in A under multiplication in A; a ring of A is a system in A closed under addition, multiplication and subtraction in 2. These cases are of particular importance later. Instead of the ray we shall generally use the commutative semigroup, which differs from the ray in that reciprocals do not necessarily exist in the system, although cancellation of common factors from equal products is legitimate. For brevity we confine the following to the case corresponding to A; the postulates are practically those of Dickson (Zransactions, vol. 6, 1905, pp. 205-208). A semigroup & is a system consisting of a set 3; of elements a, B,---,7,--- and an operation P which may be performed upon any two distinct or identical elements «, 8 of =X, in this order, to produce a uniquely determined element Pg («, 8) of =, such that the postulates (3.1)-(3.3) are satisfied. P is called multiplication, Pg(e, 8) the product of «, 8; elements of I, are called elements of ©. (3.1) If «, 8, y are any elements of & then Pa(Ps(e, 8), 7) = Pale, Pe(8, 7). (3.2) If a, 8, ry are elements of & such that Ps(e, 8) = Pole, y), then # = 7. (3.3) If «,B,r are elements of ®& such that Pg (8, «) = Pa(y. a), ‘then 8 = 7. It is shown by Dickson that any left unity w of multiplication, Pg (p, 0) = a for each « in @, is also a right unity, Pg (a, p) = «, and further that the: existence of w implies that not more than one reciprocal «' exists for each « in ©, viz., Pg(a,¢’) = pu for « in & has one solution « or none. Adjoining to (3.1)-(3.3) two further postulates, (3.4) La (et, 8) = Pg (8, a), (3.5) There exists in ® a unity pu; Pg (a, w) = Pep, ¢) = a, we shall call the ® satisfying (3.1)-(3.5) a commutative semi- group With unity w. VARIETIES OF ALGEBRA. 9 CHARACTERISTICS OF ALGEBRAIC ARITHMETIC, §§ 4-7 4. Instances of arithmetical theories. According to the fundamental theorem of rational arithmetic a positive integer is uniquely the product of positive primes; a prime is the product of no two integers both different from units. Similar definitions hold in the theory of algebraic numbers where, however, the fundamental theorem fails and where also it is necessary to distinguish between primes in a ring and elements of the ring indecomposable with respect to multiplication. It will be sufficient here merely to recall . a few of the principal definitions in order to lend reasonableness to our subsequent description of algebraic arithmetic. If «, 8 are any elements of a ring R of algebraic numbers, and if the G.C.D.* of «, 8 exists and is in R, then RN is said (by J. Konig, Algebraische Grifien, Leipzig, 1903) to be a complete holoid domain. Let R be such. Elements of R differing only by unitt factors are called equivalent. An decomposable element y of R has no factor in R not equivalent to 7 or to the absolute unit 1: if 7 is in R and is such that = divides a product «8 of two elements «, 8 in R only when = divides at least one of «, 8, = is called prime in R. An indecomposable element in R is not in general prime in R. The fundamental theorem (unique factorization into primes) holds for finite products in R. An algebraic integer is a root of an algebraic equation with rational integer coefficients, that of the highest power of the unknown being unity. The rational integers are there- fore algebraic, but the failure in general of the fundamental theorem shows that the generalization implied in this remark * The G. C.D. is defined thus, and an abstractly identical definition is assumed in any variety where the G.C.D. is significant: If ¢ divides « and fg, and if J contains every y which divides « and g, then J is called the G.C.D. of « and 8. Likewise for the L.C.M.: If x is a multiple of « and #, and if w is contained in every ¢ which is a multiple of « and 8, then w is called the L.C.M. of « and 3. TA unit in R is an element of R which divides every element of R, the quotient being also in R. Similarly, when significant, for units in any variety. 10 : ALGEBRAIC ARITHMETIC. is but partial; the strictly arithmetical character of the theory is attained only by passing to elements (ideals in Dedekind’s theory, ideal numbers in the theories of Zolotareff, Priifer, v. Neumann and others) beyond the original data (algebraic integers). In this respect the multiplicative theory of algebraic integers is abstractly identical with no part of classical rational arithmetic, for in the latter the fundamental theorem subsists for the original integral elements themselves. If however the rational integers be replaced by the principal ideals which they define (in rational arithmetic), abstract identity, for multiplicative arithmetie only, is achieved. The above remarks are intended merely to suggest the difficulties inherent in an attempt to state inclusively the essential distinction between an algebraic theory and one that may properly, in accordance with accepted instances, be called arithmetical. As it is one of the major projects of algebraic arithmetic to discover abstract identities between rational arithmetic and arithmetic in given instances of varieties, we shall attempt next to frame a definition of arithmetic which shall preserve the characteristic features of the classical theories just described and be applicable to several extensive tracts of the theory of the rational and algebraic numbers. The following may be regarded as a tentative first approximation to a postulation of arithmetic. Note that all postulates in what precedes (and the like applies to subsequent varieties) have been so framed as to exclude divisors of irregular elements, and in particular divisors of zero. 5. Restricted and complete arithmetical theories. Let & be a commutative semigroup with unity pw. If w has divisors in & (= elements of & whose product is w), they will be called units; w in any case is included in the units. Elements of ®& differing only by unit factors are equivalent; in what follows equivalent elements are regarded as identical. Then, if and only if there exists in & a subset = of elements which is such that no element of I is the product of two elements of X both different from units, and each element VARIETIES OF ALGEBRA. 11 of & except wu that is not in I is the product of elements of Xin one way only (apart from permutations of the factors), and further & contains at least one element other than wu, we shall call & an arithmetical semigroup. We say that is proper or improper according as the number of factors in the unique multiplicative decomposition of elements of X is or is not finite for each element of 3. In any instance ©; of & we write 5; for J, and g; for ». It need not be discussed here whether improper &’s exist. Let 2; be an instance of A. Then, if and only if it be possible to segregate from all the elements of 2; a ring KR; from whose elements can be constructed a set ©; of functions forming an arithmetical semigroup ©;, shall we say that 2; has (with respect to ©) a restricted arithmetical theory; it in addition the elements of &; form a ring, we shall say that A; has with respect to &; a complete arithmetical theory. And in each case the theory is proper or improper with ®;. “Arithmetical theory”, unless otherwise stated shall mean proper (restricted or complete) arithmetical theory. Similarly, if in the above 2;, 2A be replaced by any instance B; of any variety B for which the concepts of ring and semigroup are significant, including the case B — R, B; =— Rj, we define the two species of arithmetical theories for these varieties. In these definitions function is to be understood in its widest sense: if x, y are elements of any kind such that y is known when x is assigned, y is called a function of x. - For example, to illustrate an instance much used presently, if the 2; are any elements of B, and z is the matrix (x, x, ---, 24), then the matrix y = (za, 2p, +++, 7), where a, b, ---, ¢ are definite integers chosen from the set 1, 2, ..., un, is a function of x. Write for a moment aj, mj, s; for addition, multiplication and subtraction in R; = (a; m; sj), and d; for division in A. Then A; = (a; m; d; sj), while multiplication m} in ©; is in general distinct from mm. To illustrate the last remark, in rational arithmetic we have m;, mj identical, R; being here the ring of all rational 12 ALGEBRAIC ARITHMETIC. integers and =I; the set of all rational primes; &;, = RR, uw; = 1. In the theory of algebraic numbers R; is the ring of all algebraic integers in a given algebraic number field 2(;, while the elements (= functions) in &; are the ideals con- structed from elements of R;; m; is now multiplication of algebraic integers, mj in &; is multiplication of ideals, and Sis the set of prime ideals, fo; = the unit ideal. Hence, in the sense defined above, Dedekind’s theory is restricted; rational arithmetic and the revised theories of algebraic numbers due to Priifer (Mathematische Annalen, vol. 94, 1925-6) and v. Neumann are instances of complete arith- metical theories. Our definitions are therefore not vacuous. 6. Additive and multiplicative arithmetic. Complete arithmetical theories are rare, as arithmetic has developed historically into two comparatively immiscible parts, the additive and the multiplicative. This separation extends to algebraic arithmetic. In additive arithmetic we are concerned with those properties of numbers which cluster about addition, in multiplicative with those springing from multiplication, primality, and the unique factorization law. For additive arithmetic, in ., J, the appropriate smalyitd) machinery is the theory of power series of a single variable, for multiplicative the theory of Dirichlet series of one dimension—the common species. Each of these is susceptible of an n-fold gonenlinaiion, giving the corresponding theories for sets of n — 1 integers. It will be sufficient to develop the theory for n = 1, as the theory for » 1 can be placed in (1,1) correspondence with that for » = 1 by a well known device due to Gauss and used by Kronecker and others in the theory of algebraic numbers to pass from polynomials in 1 indeterminate to the like for several. The associated algebraic varieties are the module and the ring for additive, and the ray or the semigroup, preferably the latter, for multiplicative arithmetic. These also have n-fold generalizations, based on the like for 1,, but we shall attend only to the case n = 1. In the analytical theory of numbers convergence and limiting processes are central; in algebraic arithmetic infinite processes VARIETIES OF ALGEBRA. i3 in the usual sense enter only incidentally and can always be replaced by elementary algorithms. Hence we shall replace the entire algebra of power series in §F., Fr by a variety €, a special J%, whose elements are matrices (one-rowed in the case discussed here) of infinite order, and similarly for ® and Dirichlet multiplication. Both € and D are very special instances of what we shall call a matric field, which also includes many further varieties useful in algebraic arithmetic, and this itself is an instance of JF. By this means we put transcendental algebraic arithmetic on a sound abstract basis. When infinite processes are used in the sequel as in ordinary analysis, convergence of course is essential, and if it is only occasionally mentioned where relevant, for example in the theta and elliptic expansions, this is because all the processes occurring are either known to be convergent or may be shown to be so by simple means. From €, D will be constructed in a subsequent chapter a more recondite variety €, the algebra of primality and unique factorization, which yields for any 3% and for any instance of A a complete arithmetical theory of considerable interest. Both € and © illustrate Wedderburn’s theory of algebras lacking a finite basis (Transactions, vol. 26, 1924, pp. 395-426); © is apparently of a different species. 7. Crosses and possible generalizations. Attempts to cross fertilize the additive and multiplicative sections of arithmetic invariably lead to serious difficulties. Thus the classic theory of partitions is additive; one of its earliest hybrids is the arithmetic theory of forms, including Waring’s theorem; another is Goldbach’s conjecture. Again, no satis- factory definition (unless it may be in the recent attempts of Priifer and v. Neumann) has been evolved for the addition of ideals. There is however one striking exception in this discouraging prospect. The theory of the multiply periodic functions pro- vides an inexhaustible store of additive-multiplicative properties of the rational integers. Nor is this limited, as in the classical developments of this branch of arithmetic, to diophantine 14 ALGEBRAIC ARITHMETIC. equations and inequalities of the second degree. An extensive class of interesting problems of any degree may be investigated by the general arithmetical formulas furnished by the theta functions of p = 1 variables and their quotients. For this a variety PB, discussed in detail in the next chapter, emerges as fundamental. In the elaboration of L several instances of the varieties already defined appear as useful accessories. For all of €, D, €, BP and a fifth instance B of JF, useful in the study of sequences of numbers or functions, certain preliminaries concerning one-rowed matrices (or vectors) are indispensable. Before proceeding to these we may point out three ways in which the concept of an arithmetical theory admits of generalization. The first we have already mentioned: all that follows can be recast with n-fold (n >1) series and products as a background; this leads to the n-fold general- izations of the varities used here, and can be placed in (1,1) correspondence with them. Next, if we regard unicity of decomposition, the evident aspect of atomicity, as it were, as the essence of arithmetic, we are not confined to multipli- cation and its consequent & in the definitions of indecompos- able and primes, but may base an arithmetic on addition, or on any of the operations indicated presently. At least one instance of an arithmetical theory founded on primality with respect to addition exists, namely in Priifer’s theory of ideal numbers (not ideals in the usual sense), which possess a unique additive decomposition. Finally there is the possible extension of all that precedes to m-adic relations, n 2, of which a specific example is the multiplicative theory of matrices in space of » dimensions. In any of the foregoing instances there are the further opportunities of developing the arithmetic (as defined here) of the varieties obtained by modifying the postulate systems of any given varieties, e. g., by suppressing the commutative law. As a clue to an implicit arithmetic any quality of uniqueness is worth following and elaborating. This yields for example, the germ of an arithmetic of geometry, in which VARIETIES OF ALGEBRA. 15 uniqueness resides in the determination of one class by two or more classes. We return to this in the final chapter. ONE-ROWED MATRICES, §§ 8-9 8. C, D matrices, scalars, products, functions in . For subsequent use we need a few simple concepts concerning one-rowed matrices. As most of these are not current they demand detailed definition. The following is continued in § 24. A set is a collection of elements without reference to arrangement; if the collection be arranged according -to any law it becomes a one-rowed matrix, in which any given order of the elements may be taken as normal. Two normal types will be considered: the ; in which the suffixes of the elements ron 0,1, ---, and the 7), in which they run 1, 2, ~~, Thus loo, 20, "= 2a) 18.8 C matrix. of ovder n-1, (5, 5, , 2) a D matrix of order n. The C, D types refer respectively to additive and multiplicative arithmetic; when it is unnecessary to draw a distinction either the C or the DD may be used. If all the elements of a matrix are in a given variety 8, the matrix is said to be in B. I z, 2, ---, 2, is any subset of the elements of any matrix, the normal order is that in which the suffixes are in ascending order. Thus if a < ty Play, Tal| is called the D associated function (in U,,) of the element a (of D, Un) with the parameter t. Making the indicated changes from D to C notation we define gc {a, t}. If » is finite an instance of ¢y {a, t} in F, is a polynomial in an nth root of unity; if » is infinite an instance in F, is an infinite series in one complex variable; other instances in a Galois field (n finite) are evident. The following developments are of extreme generality; they provide a basis for the construction of an unlimited number VARIETIES OF ALGEBRA. 1 of theories arithmetical in the sense already described. Their important significance for infinite processes in 2, and hence in its instances §; (j = ¢, ») will be pointed out in § 15. 14. The functional varieties 2,0, * = €, 2D). The set Sx,¢ of all X associated functions gx {w,t} of the elements w of Xn Wim having the parameter t is an instance Xn.o Won of Won umder the postulates (14.1)-(14.5), in which addition, multiplication are indicated by Sx,q¢, Px. (14.1) The elements of X, ¢ WU, are identical with those of 2x, p- (14.2) Two elements gy {a, ], 9x {b, t} are equal in X, 4 U,, ¢x{a, t} = 9x {b,t}, when and only when a = b in X,U,, and hence also when and only when coefficients of like para- meters z; are equal. (14.3) The zero, unity in‘ ¥, , ll, are vx {zt}, 9x {u, t}, where z, uw are the zero, unity in X,U,,. (14.4) The irregular elements of ¥, 1, are the vx 1s, t}, where j runs through all irregular elements of %,, U,,. (14.5) The sum, product, difference of any two elements 9x ia, t}, 9x{b, t} of X, ¢ Un, and the quotient of yx {a, t} by ‘any regular element 9x {c, t}, are identical respectively with the elements gy {s, t}, 9x {p, t}, 9x{d, t}, and vx{q, t}, where s, p, d are respectively the sum, product, difference in Xn Um of a, b, and q is the quotient in X, U,, of a by b. From the last therefore, Sy, p{9x {a, tl, 9x {b, th Gr yx | Sx{a, b}, th, Prglvxia 8}, ox{b 0} = ox| Pra, b), 1); the quotient gy {q, #} is defined by Py.q {9x {4 tl, ox{c tH == wpypln tl, Pols, d =u, and similarly for the difference on replacing P by S and suppressing the restriction that 9x {c, t} be regular. That Xu, Us is indeed an instance of U,, is evident. 15. Triple isomorphism of Woy Bo Win, Xo Be IF (15) Zla,’D, on, ¢} w= 929 ALGEBRAIC ARITHMETIC. 1s an identity in any one of the varieties Wy, Xu Um, Xp, p Un, 3 denoting the zero in that variety, it is also an identity in each of the others. It is implied in this statement that ir- regular elements do not occur as divisors. This is obvious, since each variety is an instance of U,,. Moreover if (15) be an identity in Uy, its instance in %, Un can be inferred from that in Xn o Wn by equating coefficients of like parameters tj in the latter, and applying to the result the definition of equality of matrices. The effect of the last is to replace operations upon X matrices by abstractly identical operations in X, 1, upon their respective X associated functions. Since 2 is an instance of Wy, and §;(j = ¢, ) are instances of 2, the theorems apply in particular to §F;(j = ¢,r). When » = oo the instances in §;(j = ¢, ») of the theorem are equivalent to the state- ment of the necessary and sufficient conditions that those operations upon infinite series commonly called formal (but preferably algebraic) shall be legitimate and hence lead to correct conclusions. When the elements of the instance of ,, concerned are neither real nor complex numbers the theorem (when n = 0) defines the use of infinite series which have no numerical significance and for which convergence is there- fore without meaning. The numerous algorithms to which this theorem leads are powerful instruments of unification and generalization in algebraic arithmetic. 16. The associated semigroup &, of 1... Let a= (oy, --, ay), b=(8,, --., fs) be any elements of D,U,, and ¢ = (zr, - - +, vz) the parameter of any element ¢,, {a, t} of Dp op Um, so that 7; = the unity » in WU, in which addition, multiplication are indicated by S, P as always. To simplify the printing put for a moment my; = pu, ©(j) = vj, eN=o, fl =4(~—1,--., 9). Suppose now that the z;(; = 1, -.-, n) form with respect to P a semigroup Gp. Then Gp has the unity = (1) and is commutative. Let all pairs (z (7), 7 (5)) of solutions in Gp of Plz(@), z(j)} = + (&), VARIETIES OF ALGEBRA. 23 for & constant, 1 < k < n, be given by 5) = Us Js) (s=1,.--, 0), and write as the definition of pi {a, 0}, pela, b) = S{Plali), BG), ++, Play), Aju} (k =1.., n), and further define se{a, b} by seit, b) = Slew, fs} Gh =1,..., 0). Then it is easily seen that 250, $n), v= iy + +5 Vn), esl G=1---, n), vy, =v, 0 =2¢{ i=2:...2), Sia.0 = 50,0, Blab) = pila, by (G=1,..., nm), is a solution z, u, Sj{a, b}, Pia, b} of the postulates (10.1)~(10.4) of § 10, and further that in order that these be also a solution of (10.B) it is sufficient that When the last condition is satisfied we shall call &p the associated semigroup of D,U,,. By changes in notation as in § 12 we define Gc. 17. Associated semigroups of ¥,1,, and their re- lated X., ,U,,. When the order = of the matrices in XU, is infinite we indicate this as above. Of all associated semi- groups of %,1U,, two are of the first importance for algebraic arithmetic. One refers to X = €, the other to ¥ = D, and these lead respectively to additive and multiplicative algebraic arithmetic. There is also an extremely useful variant of the first, which will be defined presently. Let a = (on, oy, v3, +++), b = (By, fy; fa, --+) be any two equal or distinct elements of Co Wy. Then a solution of the postulate system for Sj, P; in § 12 is 24 ALGEBRAIC ARITHMETIC. (17.1) Sia, b} = Se), Bj}, (17.2) Pi{a, b} = S|Plao, 8}, Plas, Bia), +++, Play, Ao}, for j= 0,1, 2, ..., ond the zevo 2 = (, £, L, ---), the mity » = @,£,0,0,---) where, lags in § 2), ££ » = the Zero, unity in U,,. Yet a = (eo, 00, 05, ++-), b=, Bs, Bs, --) be any two equal or distinct elements of DoW. Then a solution of the postulate system for Sj, P; in § 10 is (17.3) Sj {a,b} = {e;, 8}, (17.4) P;{a, by =S{Fi{eq, Bia}, Plea, Bja,}s ++, Plea, Bia} Jorj=1,2,3 .... where dy(k =1, ---, ¥) are all the divisors - = 0 of j (ncluding 1, 7); the zero, wnity are z=, {, L,--), u — (v, a & HS ). That these are indeed solutions can be verified by direct substitution into the postulate systems. Indicate the solutions (17.1), (17.2) and (17.3), (17.4) by Sc, Sp respectively. Consider first ©¢. Let = be a parameter in 1, (whose unity is o, and in which S, P are addition, multiplication). Write to =, n= Pls, x, ..., 7} ==1,2.:) the = in the P product 7; being repeated precisely k times, Then the elements ©; of t = (vy, 7y, 73, --.) Juwrnish an in- stance (in the case of » infinite as here) of Gc in § 16, and we have now (17.5) Ple, vj) = uy; Jor i, j any integers > 0. Hence, a= (uy, ty) tt, 2:2) is any element of €, WU, we have (17.6) cla, t} = S| Pla, 0, Ploy, 7 ’. Pla, w), Ez as the form of any element of Cow, WU, (see §§ 13-15) in the solution Sc. Applied to (17.6) the theory in §§ 13-15 VARIETIES OF ALGEBRA. 25 is the direct generalization to U,, of the algebraic theory of power series in ., reductions of all functions of elements of Cw,¢ Wn to the form (17.6) being made by repeated application of (17.5) to all P products of parameters 5 (5 =0,.1, 2, ...) oceuring in the functions. Next, consider Sp. Let now the 6; ( = 1, 2, 3, ...) be parameters in U,, and let n= php pl (0, =1, 00,9) be the resolution of #1 into a product of powers of distinct rational primes. For p; > 1 prime, and % > 0 an integer, write ol == {0 Op, 5105000 {he 6p in the last being re- peated precisely k times. For n as above let ll Vor ol k — oe Oy == Plog, pyr =» 6}, T= 6, =v. Then the elements ©; of t = (vy, ,, T3,---) give an instance (n infinite) of Gp in § 16, and if a = (ay, a3, 3, -..) is any element of Deo Win, then the form of any element ¢p {a, t} of Deo, p Wi in the solution Sp is a1.7) 9D {a, t} = 8{Plar, ma}, Plo, 6) Play, “l 5 and reductions in Deo q¢ Wu are made to the form (17.7) by means of (17.8) Plo, vj) = where i, j are any integers => 1, and ij in the last is the product of © and j (not a double suffix). Applied to §§ 13-15, (17.7) is the direct generalization to WU, of the algebra of common Dirichlet series in F.. Since in each of Xo Uy, £ = €, D, the order of each element (— matrix) is infinite, it follows that both of these varieties contain an infinity of irregular elements, since any irregular element of ,, may occupy the first k (k =1,2,...) places in a matrix. Hence each of Xe, Wm is an instance of I, and similarly therefore for each of ¥w, Wm. This J 26 ALGEBRAIC ARITHMETIC. holds wether U,, is regular or irregular, and hence in par- ticular it holds in the instances §; (j = ¢, 7) of WU. 18. Varieties 11,, over A. The entire preceding theory can be generalized by taking 0, over 2, the postulates for the italicized process being obtained by obvious slight modi- fications from the like for an algebra over a field as in Dickson, Algebras and their Arithmetics, § 4. We shall need in particular the postulates for scalar multiplication (the elements of A being the scalars) of elements of U,, with respect to 2, and we may assume these to have been stated from Dickson. If e; is in 0, and ¢ is in the instance F, of A, the scalar product of g, e; is written g «;. To illustrate the numerous possibilities we outline a specially useful generalization, By, NU, and its associated By, ¢ Un, of Cx WU, and its associated Cu, Un, by extending the latter to U,, over F,. : The elements +¢ of Bo, U, are indicated by accenting those of G. W,, thus n= (@0, 1, ++), b= (8), BL, ok and it 4 = (5),46],...) is any elemeni of B, NW, the 4(=20,1,...) are in U, the latter being U,, taken over §,. Specifically, Bo Uy is that instance of Coo WU, in which 0; = d;/j! (j =0,1,---), where d' = (8, 01,-..-) és any element of Be Wp and d = (dy, 01, ---) is any element of Co Wi; t = (v0, 71, - +) is the parameter in Bo, Wm and is identical with the parameter t in Cw ¢ Up. Addition, multiplication in Be 1, are indicated by Sz, Pg, and are defined as follows. Let a'= (ay, a1,---), 0’ = (8, 8%, +.) be any elements of Bo, U,. Their sum, Sp{a’, b'} is the element ¢' of B., W,,,, where, forj==0,1,..., d=}, 0l,:."), a = g;/j!, o;= 8a, B;}; their product Pula, VW) is the element »’ of By, U,,, where for j=0,1, .--, le ’ ! ! ] gdb, w= mi, 7; = 8G, 0) Pe, Ao}, (G1) Bf, Bi}, ++, (oj) Plas, A), VARIETIES OF ALGEBRA. 27 in which 0! = 1, and (j, ») = the coefficient of 2” in (1-} x). With these definitions it is easily seen that Be WU, and Bieo,p Ui are an instance of €, U7, Coq Ul, where I, is U,, taken over §-. When U,, is replaced by its instance J, the instance Boo ¢ Fe of the above is identical with the power- ful symbolic or umbral calculus invented by Blissard and Lucas, which is indispensable in the algebraic analysis of sequences of numbers or functions. THE IRREGULAR FIELDS 9, €, © ASSOCIATED WITH A, §§ 19-23 19. Instances of the foregoing theory for 1,. When m= 1, ,, — A, and there is but the single irregular element 0. The unity is 1 (see §1); and S, P are now in a more familiar form, Siej, 8} = «;+8;, Plej, 8} = e; 8, where «;, 8 are any elements of 2{. We shall write (19) 2.2 =19 = BCP, 80 that J) is the instance 'when m == 1 of 9, U,, and similarly for 9, A. Addition, multiplication in 9) are indicated as before, Sy, Py. As the three irregular fields (19) are of particular importance in algebraic arithmetic it will be convenient to have short summaries of them in the simplified forms consequent upon taking m = 1 in U,. These are written down immediately from the corresponding general developments (m arbitrary) in §§ 11, 12 (with n = oo therein), 17, 18 and § 15. Elements of ¥ are denoted by oy By 7) and the z, 7; are parameters in A. 20. The irregular field €. The elements of € are a= (o,, Cy, See), hes (Bo; Bi, sy, e = (voy 715 ee), tis the parameter {= (v,,%,.- Y=, 7,72, ++); the zero, unity in € are 2 =100,0,0,...), w= (1,0,0,..-); addition, multiplication are defined by Sola:b) = 3, Pola, bl =p, where 5 = (0,, 93, ++ 3), p = (my, my, - ++), and G =e; +8j, nj = efoto ait toi} ah (J ==, 1, reals 28 ALGEBRAIC ARITHMETIC. the associated function ¢,{a,?} (of a for the parameter ?) is y dn, 1 == Ya, If (20.1) I(r, Weir ms v) = is an identity in elements x, y, -.-, v of 2 then, provided no divisor be irregular, (20.1) is an identity when x,y, .--, v are interpreted as elements of €, and the € form of (20.1) can be inferred from the 2( form from the abstractly identical relation (20.2) L(gc {x, t}, Yc {, 2, sey 0 {v, t}) = 9c {2 t} in €,, oA by equating to 0 the coefficients of v*(n = 0, 1, .--). Associated functions in € may be called power series (they are such in the usual sense only when 2A is replaced by its instances Fc, Fr) 21. The irregular field ©. The clements of D are WEE (on, tn, 000), b= AB, 0,2), = 0h, Vay )sir on the parameter ¢ = (zy, 73, - --) Where ale k, : ie Th — 9p, 6 seis 0,5 7 6 = 1 the n, pj(j = 1,..., s) being asin § 17, and the 6, where q runs through all primes > 1, being independent parameters in A. The zero, unity in Dare z= (©, 0,--.), u=(,0,0,...); addition, multiplication in ® are defined by Sp{a, b} — s, Poin, b = p, where s = (5,05, +4), p = (my, my, ++), and gj a+ Bj, n= 2arBs (rs =3 7,321) (J =1,2,.., (the sum in the last refers to all pairs (r, s) of conjugate divisors 7, s of j); the associated function ¢p{a, t} =— 27 ayzy, and tnt, = 7,, for m, n any integers >0. In precisely the same way as in € with respect to (20.1), (20.2), we have an abstractly identical conclusion in © on equating coefficients of z,(w = 1,2, 22. The irregular field 8. It will be sufficient to state the algorithm of ¥B, which follows from §§ 18, 20, in relation to the associated functions yp{a, t}, a = (ay, a, +), onl, tl] == 27 a.0tinl, VARIETIES OF ALGEBRA. 29 Raise suffixes of elements of A thus «, = e«”* symbolic exponents, as in «” being lowered after completion of " operations. in B. Then, for » = 0,1, ..., j, we write ot = 205, Yer = J, apf, © where (j, 7) is as in § 18, and symbolically opin, 1] = «7 = exper, Precisely as in , it follows that exp «r-exp ft = exp (e+ B)v, the indicated multiplication on the left being in Boo, oA. If in this b = a, one of the equal umbrae 8, « is replaced by a symbol (umbra) different from «, say B, until after the completion of all reductions in Be, A. Addition in the last being defined by expartexp Br = expr, 1; = q+ (= 0,1,-..), we again have in B a conclusion abstractly identical with that in € regarding (20.1), (20.2). In 9B the associated functions ¢pf{a, t} — exp er have algebraic properties ab- stractly identical with those of the exponential function in §., and hence the derivation of relations in B is reduced to the transformation of identities between exponentials in Fe(or in A). Associated functions in B may be called ex- ponential series, and in an obvious way we define, by means of sums and differences, ete., of such functions, the circular Junctions in A. 23. Differentation and integration in ©. An identity in € may be transformed in an infinity of ways by € ope- rations to produce new identities in €. For example, from a = bin € we infer Pla, c¢} = P{b,c}. As we shall see later, in discussing €, even the most trivial € identities yield interesting and by no means obvious relations between functions of divisors. An unlimited number of such relations flow from the operations & (differentiation) and a1 (inte- 30 ALGEBRAIC ARITHMETIC. gration) in €, which are defined as follows. Take A over Sr, getting A’, €’ (see § 18), a= (ou o,.-2), d=whea, 2), o=Wld,- 3) 1 in which the «j, «j are defined by a; == (G-+-Doey {J = 0,1, .-.3; a) = 0, aj’ = @j1/j F=12 e shall write (67, 8" are operators) 3 Then w fo ad = Da, a’ = bola, a = a, 5 = p., a = 3(8" a), a = Hoa) (r == 0, 1, ce), and therefore rar) = (0a) = a. Hence, if x, y are in € (or €’), and if for the moment we indicate € operations by the same notations xz -+y, xy, ete., as those for the abstractly identical operations in 2, we have d(x+y) = dx+0y, dry) = 23y+-ydax, 8 (xly) = (yBx—xdy)y’, Ba” == mology, and 8(cx) = c8x, where c¢ is scalar, precisely as in the calculus in §. It is unnecessary to verify these as they are implied by § 20. As a frequently useful consequence, (20.2) may be differentiated or integrated with respect to ¢ as if it were an analytic relation between convergent power series capable of termwise differentiation and integration. THE PARTITIONS OF A MATRIX, §§ 24-26 24. Coprime matrices, conjoints. For immediate use in PB, discussed in the next chapter, we continue with certain properties of matrices as defined in §§ 8-9. The elements (matrices) of a set of matric variables having no variable (scalar) in common are said to be coprime. If & 4 are co- prime matric variables, and ¢ is a scalar, c¢&, cg are coprime. From the » independent variables xz; of § = (zy, ---, Zn) can be constructed precisely 2”—1 matric variables of orders VARIETIES OF ALGEBRA. 31 < n (since by §§ 8-9 it is presupposed that all matrices are normal). Let & (j = 1, ..., 2*—1) be these matric variables, and let the ¢, ¢; denote scalar constants 3 0; also let &,, &, ---, § be coprime elements of the set & (G=1,..., 2»—1), all of whose elements together are the set 2; (6 = 1, -.., wn) of the elements of Z. Then § is called the conjoint of &, &;, -.., § and we write F = f+ Ett, The conjoint 5 of c,&,, csp, +--+, ¢,&, Is n= bole egdg+---+c, §, which is defined by the preceding since the ¢y&; (0 =a, 8, +--+, 7) are a coprime set. Non-coprime matrices cannot be conjoined. Conjunction, which in * replaces the usual addition of matrices in 2, is analogous to logical addition; it is commutative and associative. The conjoint of —c,&,, — cps, ++, —e,&, is —5. When convenient conjoints are enclosed in parentheses, gy = inf. t oxfs tr. a dl), and evidently such expressions obey laws abstractly identical with those for the like in A. Thus, for example, §, 5, 6 being coprime, we have wf byd fd == or (Bop—A) mm Er pd), The zero conjoint (0) (corresponding to the null class) is the matrix having no elements; (0) 4 (0). The matric product of any number of matric variables is the matric variable whose elements are all the variables common to all the factors; the matric sum is the matric variable consisting of all the variables (each taken once only) in all the factors. These definitions are included only for completeness; they are not required in PB but are useful in the theory of arithmetical structure (which will be briefly sketched in the concluding chapter). 32 ALGEBRAIC ARITHMETIC. 25. Residues in a.module. Leto (= 1,2,-..) be elements of a module MM. Then, 0 being in M, a;— a; = 0 = 0w; = 0 for each «; in WM, and any element a $0 of IM is of the form og — kooeq + kyo + - . kee, where the %’s are rational integers + 0 and the «’s are distinct. Replace each %; by its least positive residue modm; call the resulting element of IN the positive residue of e¢ modm, and write a =— mye, + mya +--+ mea, modm. Each element of 2t has precisely one positive residue mod m. The set IM, of all positive residues mod m of elements of IM is a module, say M,. The sum (difference) of any two elements of Mi, is the positive residue mod m of the algebraic sum (difference) of the given elements. The last defines M,.. in which each element « is of the form rq e+ rpep +--+ ree, where «,, a, ---, a, are s distinct elements of I, and each of vq, 74, +++, rc is a definite one of 1,2,..., m—1. Hence it MM has the finite basis [ey, ---, a], M,, contains precisely m*—1 elements § 0. : Let each ¢; (j = 1, 2,.--, un) be a definite one of 1, —1. Then, of the 2% distinct elements of the form CaVa at eprpoty +--+ ero of IM, where the e's take all their possible values, precisely one, namely &, given hy ¢,==g = .-:==¢, = 11s in W,. The remaining 2°5—1 elements of this form are called the conjugates in M of «. Any element of IM, is the positive residue of each of its conjugates in 9M. The element a +oyt-.-+ a, of M,, whose basis is [e;, ap, .-- a] Is called the trace of Mn. When m = 2, the case applicable to PB, each element «30 of IM, is of the form 4+ apt. - + a, where the «’s are distinct, and the conjugates of « in IM are eaatepop t-te, (p= 1,7 = gba. VARIETIES OF ALGEBRA. - 33 As m = 2 is the only instance which will be applied in detail we confine the following remarks on partitions to it. 26. Partitions in I,. Let 1; (j=1,---,n) be n coprime matric variables. Then [w1,us,---, uy] is the basis of an instance IM: of M., whose trace wp is the conjoint pg + pet + wy; the conjugates of w are the ey ps +e ps4. +e, uy, (ej = 21, J=1,-.., nm), with the exception of u itself. Except in the case (trivial in applications) n = 1, w = a scalar variable, x can be separated into a set of conjoints in more than one way; call any such set a partition of pw, and let mw = po+ pp +--+ + pe be any partition of w. All the m are obtained by distributing all the elements of @ in all possible ways into coprime sets, forming from these sets (normal) matrices, and taking the conjoint of each resulting set of coprime matric variables. If the order of win [pier ul is op (5==1,..., 7m), 5 in of order ® = w+ --.+ o,, and the total number of partitions = is the number /7(w) of distributions of w different things into non-overlapping parcels—a well known function in combinatory analysis for which there is no concise or usable expression. This lack is unfortunate, as /7(w), we shall see, is in a precise way an index of the generality of the general arithmetical theorems implicit in an identity between elliptic and theta functions involving precisely w independent variables. The set of all partitions 7 together with their conjugates are fundamental in the applications of periodic functions to algebraic arithmetic, and these applications are themselves instances of the algebra 3 considered next. They also appear as basic in the applications of 3, particularly to the Bernoullian numbers and functions and their generalizations to allied functions of » complex variables, but we shall not have space to go into these in detail. CHAPTER II THE ALGEBRA PB OF PARITY ABSOLUTE AND RELATIVE PARITY, §§ 1-6 1. Origin of PB. Functions admitting expansions into sums of powers of linear homogeneous functions of their arguments, also certain other fypes (in a technical sense) of functions to be noticed in the next chapter, give rise in their applications to algebraic arithmetic to the interesting algebra P of parity. Conversely, for the full and efficient develop- ment of the algebraic arithmetic implicit in the analytic theory of such functions, for example the elliptic and the theta of »=>1 arguments, P is essential. We shall therefore consider it in some detail. By itself 8 is an interesting example of the abstract identity of the simultaneous solutions of several overlapping systems of postulates. In the following presentation of the main outlines of. the reasoning is necessarily of a somewhat abstract character, as the 3 theorems relate to functions which are entirely arbitrary except in two respects: each function has parity, as defined presently, and each takes a single definite value for each set of integral values of all its arguments. Hence in par- ticular all assumptions on the final functions as to con- tinuity, differentiability, or expansibility into series of any type whatever — in fact all the customary machinery of analysis except uniformity with respect to integral argu- ments — must be avoided in the proofs. The point con- cerning assumptions as to expansibility, particularly into trigonometric series, is to be specially noticed, as it is by such excluded means that an exceedingly simple and equally fallacious method of “proving” certain of the principal theo- rems in their unrestricted forms is at once suggested. This very absence of restrictions upon the functions in P is the 34 THE ALGEBRA } OF PARITY. 35 essential and sometimes rather elusive crux in the proofs of those theorems concerning the functions which are most fre- quently used in subsequent applications to algebraic arith- metic. As the theorems of P are ultimately finite identities be- tween matric variables whose elements are rational integers, we shall avoid so far as is feasible without undue elabo- ration all reference to infinite processes, although certain of the basic lemmas for specific applications, for example those connecting PB with elliptic functions, can be obtained by such means. The final arithmetic being algebraic and not analytic, it is fitting that a minimum of analysis be em- ployed in obtaining the fundamental theorems. In the last’ step connecting P with the algebraic arithmetic of the common periodic functions it is sufficient (but not necessary) to assume only the power series for the sine and cosine. The complete PB, abstract structure and instances, is a novel example of the application of algebra to analysis with the object of obtaining applications of analysis to arith- metic; the algebra in turn can be replaced by elementary identities in rational arithmetic, so that in the end we have applications of rational arithmetic to analysis instead of the more usual reverse. In following the development of P with a view to possible generalizations, some may be interested in observing that the elemental, generative fact underlying the entire theory of parity in all of its ramifications—which are many—is the protean one that any rational integer when divided by 2 yields one or other of the positive remainders 0, 1. Hence, since the theory of elliptic functions is contained as an instance in some of the simpler identities in $5, all of which can be proved independently by elementary arithmetic, it follows that double periodicity, ete., can be traced to the same simple source. There is a generalization of to moduli »>2, and the like applies to ® of the preceding chapter, in which the units of any algebraic number field replace the units --1 of rational arithmetic in the case n>>2, and there is a further git 36 ; ALGEBRAIC ARITHMETIC. generalization with respect to any algebraic number field. The analytic functions to which the last extension could be applied with profit to arithmetic have yet to be investigated for PB; they can be easily constructed in the form appropriate for B. Hecke’s theta formula is an instance of the type of theorem which can be applied to an extended 3. A great desideratum for 9 is a practicable form of the trigonometric series for the n-fold periodic functions, »>2. The discussion by Appell (Acta Mathematica, vol. 13, pp. 1-174), for n =4, analytically complete, abandons the problem precisely where its utility for arithmetic begins. 2. Absolute and relative parity. The definitions already stated for matrices, their functions and partitions, are pre-- ~ supposed. We shall develop ¥ with respect to A. Hence ‘all matric variables are in A. Parity is the eveness or oddness of a function with respect to its matric variables. A function whose value remains unchanged when the matric variable z of order n is replaced by —z is said to be of even absolute parity nm in z; if the value of the function changes sign when z is replaced by —z, the function is of odd absolute parity nm in z. The even, odd absolute parities just defined are written p(n 0), p(0 nr) respectively, and the absolute parity of a constant with respect to z is p (00). The absolute parity of a function is the same as that of any of its constant scalar multiples. An arbitrary function f(z) of z has in general no parity. Such a function however may be written as the sum of two functions of the respective parities p(n|0), p(0 un), where n is the order of z, 27) = l/r 7 [ fi) —F(—2)], since f(z) =f (—=z) are of these respective parities, precisely as in the common instance of this for n = 1. Let 4;, i=1,--v,7; §=1,...,8 be 7-}s coprime matric variables of the respective orders a;, b;. Then a function / whose even absolute parity in 4; is p (a; | 0) THE ALGEBRA 3 OF PARITY. 37 (¢=1,---, 7), and whose odd absolute parity in w; is p (0 |;) (j=1,.-.,s) is defined to have the absolute parity (2.1) ln, wo, in, rv, Di) When necessary to refer to the matric variables 4;, uj, and not merely to their orders a;, ;, ‘we shall say that f has the relative parity (2.2) Pay -ey dp pu, ee, ps). An absolute parity is a function of positive integers, a rela- tive parity is a function of matric variables. If the odd obsolute parities are lacking, (2.1), (2.2) will be written (2.3) p (ay, Pty ar | 0), p(y, ed Ay 15 similarly, if the even absolute parities are absent we write (2.4) (0 | by, Text bs), p( His = 2% Hs). Unless otherwise evident from the context p is used exclu- sively to indicate parity. Any function f of the matric variables 4; w; having the parity (2.1) or (2.2) will be written (2.5) JS (A, * ay Ar | py, Ty ts) ; corresponding to (2.3) we write in the same way (2.6) g (Ay, ty Ar D, and to (2.4), 2.7) h(| py, ++, ps). In the symbols (2.1)-(2.7) the arrangement of the a; bj, Ay 15 is immaterial provided only that no letter passes the bar. If in (2.1) precisely « of the a; G = 1, ..., 7) each = a, and precisely 8 of the o; (j = 1, ..., 5) each = ¥, we replace all these equal a; or 0; by af, b% in (2.1) and, pro- ceeding thus, define 38 ALGEBRAIC ARITHMETIC. pla”, b? Tey +f | 2, ih 22 £), in which a, b, ---, c¢ are all distinct and likewise for », s, ---, ¢. In no symbol of a relative parity or of a function having parity such as (2.2)-(2.7) can any matric variable appear twice, since the 4; wu; were taken coprime in the initial de- finition. To avoid a possible confusion we recall that coprimality of matrices was defined for matric variables, not for their values. Thus, for example, if x, y, z, w are independent vari- ables, the matric variables (x, v), (z, w) are coprime, although they may have an infinity of equal values (er, 8) ¢ = 1, 2, ...). A proposition concerning either variables or matric variables implies instances concerning values, but not conversely unless the instances refer to the set of all possible values. The latter possibility will be discussed in connection with the theory of division of parities. In the matric variables 4; wu; the functions f, ¢, & in (2.5)-(2.7) have by definition the absolute parities (2.8) p10, (17100, pU0[19), the double (()) being used to distinguish these from the ab- solute parities (2.9) pi), (lO), 2001) obtained from (2.1) when respectively m=1 h=1: m=1 b=0 ao oh ="1, G=1,...,r; j=1, 8) It is sometimes convenient to include » — 0 or s = 0 in p(17] 1%). We assign by convention the following meanings, p(1°| 19 =p] 19), pA" | 1°) = p(17| 0). Similarly for (2.8). Thus (2.9) is the special case of (2.8) in which the 4;, u; G=1,....7: j=1,.--,9) are all of order 1, and p17} 19) is the absolute parity of a function of precisely » +s in- dependent variables, even in each of » of them and odd in the rest; p(17|0) is the absolute parity of a function of THE ALGEBRA 5 OF PARITY. 39 precisely » independent variables, even in each; p(0| 1%) is the absolute parity of a function of precisely s independent variables, odd in each. Again, p(r| 0) is the absolute parity of a function of precisely » independent variables, even in all » simultaneously, and similarly for p(0|s) and an s-fold odd function; in the matric variables these have the re- spective absolute parities p((10)), p((0 | 1)). An extension of these concepts is important shortly. Let’ Ai, pj in (2.5) be partitioned into aj, b} matric variables (¢=1,.---,7;j = 1,..-,5). Then the absolute parities of Sig, in (25-2. inthe gy +--+ a. +0. 4... 4 0} matric variables occuring in the partitions will be indicated thus, p (a1, aT a | by, RR bs), p (a1, Nis ar | 0), p((0 | bi, sary bs). This merely defines the symbols (2.10); their Tin will appear as we proceed. When mi =a, Ui =U G=1,..., 7; f=11 29) (2.10) becomes identical with {2.1} i its rat As an example of (2.10), let 2, w, » be coprime matric variables of the respective orders 7, m,n. Then p((2]1)) is the absolute parity of f(A wu |») in 4, gw, », while in the variables (—— scalar elements of 4, w, ») bio absolute parity is p(I+ m | n). 3. Absolute parity of a relative parity. As before let 4; wu; be coprime matric variables of the respective orders ai, bj and let the uw; v; be constant scalars. Then by § 2, (2.10) Jl, A Ar | ws, rent Ms) s Ju dy, ty Wrdy | vip, ht Us 0s) have the same absolute parity stated in (2.1); in the matric variables they have the same absolute parity p((17]1%)). Considering the relative parity Dp (1 44, Si Ur hy | Villyy >: 7, Vs Ws) of the second as a function of the matric variables we assign to it the absolute parity p((17|1%)). This accords with § 2. In partienlar then, for 4 = 1,..., 0, j =1,0.0 2 40 ALGEBRAIC ARITHMETIC. p(y, ery, —Ai, ee 2, pay, very Wg) = p(y, suey dit, ‘es Zo fey von tis)s py, Ap | fay omy — jy ony fs) —pldy, ks Al pyy eo, 1, fee Ms). 4.. Order, degree of a function having parity: These are fundamental in 9B and in its applications, particularly to the elliptic and theta functions. Referring to (2.5) we write now 0p = 7, 0, = gs 0 = 0+, and call these the even degree, the odd degree, and the degree respectively of f; we also define 7 Ss 2 wy, = NN. 0, = bis w= wy} vo, =1 J=1 to be the even order, the odd order, and the order respectively of f. Similarly, with obvious modifications for g¢, 2 in (2.6), (2.7); for example, the even order of % is zero. 5. Statement of two theorems. The import of the preceding definitions will be plain from two theorems in PB which will later be obtained as very special cases of a general theorem which it is the object of the algebras I, Ry, in- troduced in a moment, to derive and reduce to a simple algorithm. The following will be recognized as further ex- tensions of the familiar theorem which expresses a function of one variable as the sum of an even and an odd function— which we have already extended in one direction to n>1 variables. . (5.1) An arbitrary function of o independent variables is the sum of 2% functions having absolute parities of the forms p(1%[1%), where a+b = wo, and the complete statement of this which gives the number of functions of the fixed parity p(1*|1Y), a, b constant, in the sum, is abstractly identical with De Moivre’s theorem in Fe. : This follows from another for which we shall have more frequent use. (5.2) Any function having the absolute parity p(as,,ay, by, bs), of order ® and degree 9, is the sum of 2°° properly chosen THE ALGEBRA ! OF PARITY. 41 Junctions whose absolute parities are all of the type p (1 17), where a-+b = w, and the complete statement of this which gives the sum is abstractly identical with the formulas in $F. Jor the decomposition of a product of r cosines and s sines mto a sum, and the addition theorems for o arguments of the sine or cosine according as s is odd or even. The generalization of (5.2) refers to functions having the absolute parity p((a1,---, a, | bi, ---, bs)) and becomes identical with B.2) when t= wy, W; =U; G=1,..:, 7; 7=1, i. 5). By considering the set of all partitions of the trace ¥ EE dyrh ere cb dp bn hos ag, and applying the generalized (5.2) to f(»|), (|v), we shall later obtain all the general arithmetical formulas (involving . functions arbitrary beyond their parities) implicit in any identity between elliptic and theta functions, or either alone, in which precisely ® independent variables are involved, where o is the order of ». Thus the consequences of what is next developed are far reaching. Corresponding to the generalization of (5.2) there is an immediate extension of (5.1) to entirely arbitrary functions expressed as sums of functions having parities p ((1*/1%)), but as this is less useful than the others we shall omit it. 6. The semigroup of coprime relative parities. The members of a set of relative parities having no matric variable in common are called coprime; a matric variable and its constant scalar multiples are not distinguished in this definition. Thus p (Z|), p(|—2) are not coprime. Multiplication for non- coprime parities is not defined. Let f, g, 1 be functions arbitrary beyond the respective relative parities implied by their notations, F=00a,---, |p, --, ps), gy = gles, 0tl01,---, 0), h = h(21, “rey dip, 01, ee 0m, rity fin, Of, rs, Ou), where Ai, Sy Ars Hiy 2% Hay Quy wire Oty Op, vv, Gy ATC r+s-+t-+ wu coprime matric variables. Then obviously 7% 42 : ALGEBRAIC ARITHMETIC. and fg have the same absolute parity. The product of the relative parities of f, g is now defined to be the relative parity of &. Indicating this multiplication by juxtaposition we write (da, sats Ar | 1, sey fs) p (01, s Sih, ot 01. vey Oy) he pd, ey Ar O01, +, ot | pr, ~rcy Mgy O01, :--, a,), valid for any » + s+ ¢-}-u coprime matric variables Z;, tt, 05, 07. The arrangement of the matric variables within a symbol p being immaterial provided only that the bar be not crossed, it follows that multiplication of relative parities is associative and commutative. This multiplication has neither unity nor inverse. It is clear however that with respect to multiplication the set of all coprime relative parities formed from a given set of matric variables is a semigroup. The notation being as above we have the following im- portant special cases, YisGh=0pc. 2.5 Tlstm == 20m -- ua, i=1 Jji=1 r Ss 112¢:D Told = 5G, --», Llu, +++, m0) i=1 J=1 In the less obvious theory of addition of relative parities constructed next, the following remarks will be frequently assumed further notice. Let p = wo + wp + «+ + pe, be any partition of the matric variable wu, and let Me = eaMa + eytp +--+ + ec pe, where each eg; (j =a, b, ---, ¢) is a definite one of 1, —1, be any conjugate of this partition. Then f(w ), f (ue) have the same absolute parity p(m | 0), where m is the order of w, and similarly for f£ (| u), £( re) and p(0|m). When gp is replaced by — pu, we becomes — ue, and hence the true equations S(—etappo—---—ep|) = fleapat --- + ecpe), Fl —eapa—-- —ecp) = —f( east ---+ cep) are implied by the single change of w into —uw. Again, illustrating for the simple case uw = wu, -+uy, we see that THE ALGEBRA 5 OF PARITY. 43 JS (w ), J (a, fo ) = 7 a, wy) have the same relative parity p(u|). For, if © be changed to — pu, f(w)) is unchanged in value, and under this change the difference becomes Fl— va; — 16 )—f( — pa, — 10), = fa, 0) —S ta; 3). Similarly for Sw), fa | wo) +f (0 | a), which have the same relative parity p(w), since Sta | —pw) + f(—pn | —pa) = —f (am) — 7 (| 1a). As another illustration, let A = 4,4 A, — 4, where 44, 4, 4, are coprime matric variables. Then f (| 4) and Ji Qo, Ae | 2a) +13 (Rey 2a | 2) =f (hay 2p | De) + fi (| hay Da, Ao), where f, fj(j = 1, -.., 4) are arbitrary beyond their indi- cated parities, have the same parity p(| 4). ABSTRACT IDENTITY OF B® WITH THE ALGEBRA OF THE CIRCULAR FUNCTIONS, §§ 7-9 7- The functions ¢; (j = 0,1). Let u,v, us 0: G=1, 2...) be elements of a module 2 in A, and let ¢; (uw) (j = 0,1) be functions forming a ring Ry as uw runs through all elements of MM. Then, the indicated additions, multiplications, sub- tractions being operations of Ry, $0 (1) = 91 (v), $0 (1) == 90 (v), 91 (1) = 9; (v), po (1) 91 (v), Po (10) 90 (v), $1 (0) 91 (v) are in Ry. The additions, subtractions in ¢;(4u), ¢; (u ==) (j = 0,1) being now in IM, these ¢; are in Ry. On the set of all ¢; we henceforth impose the postulates (1) $j (—u) = (— 1)7 gj (uw) (y = 0,1), (1.2) 91 (uv) = 91 () 9p (v) + (— 1) 9; (u) 9, (v) (j = 0,1). 44 ALGEBRAIC ARITHMETIC. From these follow, for ; = 0,1 in each instance, (1.3) gr j(u—2v) = 91,0) po (v) — (—1) 9; (0) 9; (v), and therefore by (7.2) (7.4) 291 jw) po (v) = 91; +0) + 91_j(u—0), (1.5) 2(— 1Yo;(u) 9, 0) = 91 ju 9)— pr ;lu—v). In the instance Fe of A we define T to be that part of trigonometry which, when go (uw), 9; (uw) are replaced by cos u, sin wu respectively, is implied by (7.1), (1.2) alone. When UA us abstract, as always unless otherwise specified, the set of all propositions implied by (1.1), (1.2) will be designed by Re. 8. The $ isomorph of R,. We now solve (7.1)-(7.5) in terms of relative parity. Replace u,v, up 5G =— 1,2, ..) by a set 4, w, A;, ;(¢ = 1, 2, ...) of coprime matric variables. We recall that multiplication of relative parities was defined in § 6. Let » be an arbitrary matric variable. Then (8.1) 9») = pF), nn(») = p(») 1s a solution in this instance of (7.1). Under the substitutions (8.1) we get from (7.2) (8.2) pA+u)) = p@)) plu)—p(i) p(w), (8.3) p(A+wp) = p(2) pu)+pQA)) pu), or, what is the same by § 6, (8.21) plA+p) = plu)—p(iuw, (8.31) p(A+p) = pli) +pQ lp), which as yet are without significance. If a consistent inter- pretation can be assigned to (8.21), (8. 31), we shall call (8.1) a PB isomorph of Re. : The interpretation to which we shall adhere is as follows. For (8.21): an arbitrary f(A + w|) has the same relative parity ppl) as the difference fii, p|)—fa (|, pn), where fi, fo THE ALGEBRA 8 OF PARITY. 45 are arbitrary beyond the indicated relative parities p (A, |), p(| 4 w), and similarly, mutatis mutandis, for (8.31). Further, we shall write these interpretations in the symbolic forms (8.22) SO+u)) = fO,u)—f(4 un), (8.32) J(2i4+m) = fO|n) + fled), obtained by replacing p by f. Similarly, from (7.4) we write down (omitting the intermediate p forms) (8.41) 2fG, pu) = fO+u)+f0—pn)), (8.42) 2/G | mw) = f(A+pw)—Ff(i—pn), and from (7.9), (8.51) —20 4) = JO tu—fU—n]), whose interpretations are inverse to those of (8.22), (8.32) in the following sense. Note first that on the left of (8.22), (8.32) each function is of degree 1 (since Aw is a single matric variable in each case, thus f(42-+ w|) has the absolute parity p(n |0), where n = the sum of the orders of 2, uw), while on the right each f is of degree 2; in (8.41)-(8.51) the exact reverse obtains. It will be sufficient now to in- terpret any one of (8.41)-(8.51), say the first. The constant multiplier 2 does not affect the parity of f(A, u|). The inter- pretation of (8.41) is that f(A, w|) (or 2/4, 1) has the same relative parity pA, wl) as the sum fi(A+p)+fi(h—pu)), where f, fi are arbitrary. Note that in the interpretations of (8.22)~(8.32) the f’s on the right typify different functions having the indicated parities, while in (8.41)—~(8.51) they refer to the same function. ~ To illustrate one of the important identical transformations discussed generally in a moment, apply (8.41), (8.51) to the right of (8.22). Then Slt) = Sf Ou) +f Oph] + LO uh—f ap], 46 ALGEBRAIC ARITHMETIC. which in addition to checking the formal accuracy of the algebra gives us the following: an arbitrary function f(A4 pul) having the absolute parity p(/-+m 0), where I, m are the respective orders of 4, w, is the sum of two functions of the respective absolute parities p (I, m0), p (0/1, m). This is the immediate consequence of applying to the processes yielding the above algebraic identity the appropriate inter- pretation as defined. After this detailed example there will be no dificulty in following the abstract discussion of the general case. 9. Abstract identity of I, R,, P. Let & be any relation in R, of §7 which is implied by (7.1), (7.2), and let RB become 7 in the instance I of R,. Suppose that inter- pretations can be assigned to the elements of A, giving an instance A’, such that (7.1), (7.2) are consistent in the instances IM’, Ry of M, Ry furnished by A’, and let RB when interpreted in Rf be R’. Then R’ is implied by either of &, T, and hence all the true propositions in Rj, Re can be written down from those in £. We define B to be the set of all propositions implied by (8.1)-(8.51), or what is the same by (8.21)-(8.51), together with their interpretations as given in § 8. As is well known, T is sufficient in $, for the deduction of De Moivre’s theorem for a positive integral exponent, the expansion formulas expressing the sine or cosine of the sum of » arguments as a sum of products of sines and cosines of single arguments, and the decomposition of such products into sums or differences of sines or cosines of linear homo- geneous functions of the arguments. It is precisely the ab- stractly identical equivalents in PB of these T formulas that are important for applications. We shall write them down for Ry directly from T by means of § 8 and the abstract identity noted above and then, by an application of 3, infer immediately their interpretations for functions of matric var- iables having parity. What follows is more than a set of existence theorems; it gives a short way of obtaining the actual representations proved to exist in the 9} theorems. THE ALGEBRA 8 OF PARITY. 47 EXPANSION AND DECOMPOSITION IN $B, §§10-14 10. Decomposition in Ry. For brevity we shall write 9,00) 9,00) = 9,0) = 9,005 ty sw) (G = 0, 1); the indicated multiplications are in Ry. For either value of j this defines a species of multiplication, not necessarily that of Rp, under which the set of all 9; (uy, +++, u,) (r=1,2,...), as the wu; run through all elements of I, form a commutative semigroup according to @; (uy, 23) ,) 9; {u,, ey v,) T= @; (uy, waive Ws Upp res v,) (J p= 0, 1) On this multiplication we impose the postulate Pv (20, u,) ¥; (vy, 0) = 9; (vy 0) Pj (ym, (=0,1), so that with respect to it the ¢(u;,---,u,) (r =1,2,..-; J = 0,1) form a commutative semigroup. Since Ry is a ring, this multiplication is distributive, as in 9. Let 7, s be integers 0, and let > refer to all sets of values of the ¢ each = 1 or —1. Then for 7=1 the generalizations, implied by T, of (7.4) in Ry to » arguments is (10.1) 2 go lwp, ms, + +o, 1) 52 2 walt +00 + ook 0), and that of (7.5) to 2s arguments for j = 1 is (10.2) 2 U—1¥ op, (uy, 00, =~, ss) = > aly. so (ty + eats +--+ eas ss), which is equivalent to 10.21) 22(— 1 play, 4; ++ +, Us) = eres rspoley ty + ert +--+ + eaottas); also, when ¢>1, (10.3) 22-2(—1) 1g; (uy, up, - ++, tre) = y eaCy---at—191 (Uy + aus +--+ e201 Uat—1), 48 ALGEBRAIC ARITHMETIC. and if 10 (10.4) 2% ==1)"1 Py (us, Uz, +++, Ust— 1) = pion on ole w =k opin + oso mei), Each function on the left is a product in R,, thus @o (Us, +++, 1) — (uy) ++ - 9p (uy), ete., while each right hand member is a linear homogeneous function in Ry, with co- efficients 41, of ¢, or ¢, functions, each of a single argu- ment in IM, the functions in a given instance being all ¢, or all ¢; according as the number of ¢, factors in the Ry, products on the left is even or odd. The number of terms on the right of (10.1)-(10.4) are respectively 271, 225—1 926-2 221 a remark which will be assumed in writing down in § 15 the general PB theorem mentioned in § 5. Either directly from the isomorphism with < or from (10.1)-(10.4) we have the following generalizations of (7.4), (1.5) with 7 = 0, valid for integers », 5, t > 0, (10.5) gL (=1} Po (21, Way 7° 7 uy) $1 (vy, Va, ==, Vas) = > Ev Caso (Us + Cay +++ Terttr 1&1 + vp 1... + E95 Vas), where X refers to all sets ef values of the e;, 5 (6 = 2, ..-, 7; j=... 29 each == 1. (10.6) Sohne? (1) Po (ug, Way x, Uy) 1 (v1, va, tt vat—1) = Deeg. np taut- tonto eames omy Un) These contain respectively 27+2s—1, 2r+2-2 terms, We shall refer to (10.1)-(10.6) as the decomposition formulas in Re. 11. Expansion in Ry. The presence of i = (—1)"2 in an Ny identity signifies only that the coefficients of 7, 1 are severally equal, precisely as in ., when the whole identity is reduced modulo 2+ 1. De Moivre’s theorem in $ is then abstractly identical in R to THE ALGEBRA 5 OF PARITY. : 49 (11.1 11 [po (uj) +91 (u))] = go tte, T. J=1 J=1 j=" Comparing coefficients of 1, 7 in the distributed form of (11.1) we get the following expansion formulas in Ry, (11.2) oo (uy + us +--+ +2) = 4 Aro By +X 4,4 By — des Bst:-., (11.3) $1 m+. ty) = di Ayn Bb — 4-3 Bs LY Ads By Til in which 4, ;B; is the product of »r—j functions ¢,, each with only one of up (k = 1, --., r) as argument, all the arguments being different, by ; functions ¢; whose arguments are the remaining wi, and XY; A,_; Bj is the sum of all such products for j constant. 12. The identical transformations 7, 7, in #,. To each term on the right of (10.1)-(10.6) apply the appropriate one of (11.2), (11.3) to expand. Then, directly from the abstract identity with 3, it follows that the new right hand member reduces in each instance identically to the left. This process of expanding the decomposition of a function in My will be called the identical transformation Ils. If conversely (10.1)-(10.6) be applied to decompose each product on the right of (11.2), (11.3), we get the identical trans- Jormation I; These are fundamental in 23. : 13. Expansion in B. From the abstract identity of Ry, PB in §9, and from the interpretation in § 8 of (8.22), (8.32), we can now infer the general expansion formulas in B, of which (8.22), (8.32) are the simplest instances, and infer immediately their interpretations. These and the like for decomposition can readily be verified independently by mathe- matical induction if desired, but this is superfluous. In abstract identity with (11.2), (11.3) we now have (13.1) f' (2, 4 Ly + al + Ar | ) = 4H— Arn Bs | wee, (13.2) f( | A + Ly Ie Dud +1.) Ee x, Ar—1 by — 2 Ars By + dy 4 50 ALGEBRAIC ARITHMETIC. in which 4; (0 = 1, ..., r) is a set of coprime matric variables, and where now 4, ;B; denotes a function fhaving the relative parity indicated in J(u, May > vy Ur—j| Vy, Vg +=, v;), where ppl — 1, .-., r—j) is any set of r—j matrices chosen from the set 4;(6 = 1,..-, 7), and v,(s = 1, ..., 7) are the remaining j matrices in the 2; set, and x for constant, refers to the sum of all such /’s for the »!/5! (r—))! possible choices of the py, vs. The interpretation is that 4f the fs on the right are replaced by arbitrary fi, fs. «+ having the indicated relative parities, then the arbitrary f on the left has the same relative parity as the sum on the right, which is pl) in (13.1), p(|.4) in (13.2) where, 4 =) +3 +--+ hp 14. Decomposition in Pf. From § 10 we infer, as in § 13, the decomposition formulas in PB, where r, s, ¢ are as in the correspondingly numbered formulas of § 10; the 4; yi=1,2...7 = 1,2...) are coprime matric variables, (14.1) 21700, 25, ++, 4p) = rl) + eghy +--+ 0.4, ) (14.2) 25H (—17F (pg, p, - ++, p12) my ea ey asf (py + apts + + eas pins |), (43) YF, 0s +o po) = Y essen f(t eps t- - -enrapmy), (144) 2% Y(—1¥-1f (1m, pa, - - +5 pot) = eee fet +e s+. eng tot 1), (14.5) DEI Ul V Fh dy, oe, dp | ys fas = oy fag) = oy &y-bns fh tesdot teh tp Feats + oo cet Eagptas ), (14.6) 2020 Yr QJ, on: Zo [pty 12, ++, fat—1) wm Erb fh tedyt Fedde tut. cet Ear para), of which the interpretation is as follows: ¢f the several f’s on the right of each of (14.1)-(14.6) be replaced by the same THE ALGEBRA Pi OF PARITY. 51 Ji, arbitrary beyond the parity implied in the notation, then Joon the left in each instance, arbitrary beyond the indicated parity, has the same relative parity as the sum on the right. THE IDENTICAL TRANSFORMATIONS IN $B, §§15-17 15. The general decomposition-expansion theorem in 3B. The following theorem is fundamental in the algebraic arithmetic of periodic functions. If in J =000, os hla, - +5 ps), . of absolute parity p(ay. ---, ay by. ---, bs), the matric variables Lis pt; be partitioned in any way into aj, bi new matric variables ((=1,.---,r;5=1,---,8), so that in the matric variables of all the partitions f has the absolute parity CE Ae and if 1 Ss o = d+ 2 F=rts o—8=g, = = then fis a linear homogeneous function with coefficients +1 of 2¢ properly chosen functions of all the matric variables in the partitions, and the absolute parity of each of the 2¢ functions in the matric variables of the partitions is of the form p((1¢/1%)), where a+b = o'. When as=w;, Ul== ly li==1,-+., 71 d=], .., 3), his degenerates to (5.2). The general theorem is the interpretation in P of I, 1;. -As a practicable method of obtaining the actual linear, homogeneous function described is necessary in the algebraic arithmetic of periodic functions, we devise such a method next, and this implicitly contains a fuller proof of the theorem. 16. The identical algorithm in 5. We saw that Rs TT, P are abstractly identical. Hence we may operate in any one and interpret the results in each of the others. We shall operate in the familiar T and infer ¥. As in the principle of duality in geometry a correspondence is established between 4% H2 ALGEBRAIC ARITHMETIC. the elements and operations of the two systems which makes computations necessary in only one. When interpreted in T the Greek letters shall designate independent variables in §.. when read in 3 they shall denote coprime matric variables of any orders whose elements are in A. Sums and differences of variables in T, occurring as arguments, are as in F.; in J they are conjoints, and all differences indicate conjugates of partitions of matric var- iables.. To (16.1) Fl, vo, holm, vv, pig) in Pf corresponds (16.12) COS 2; : +. €OS A, SiN py - - - Sin ps in ET, and to G62) Flnkiat-: £4], Flint tu) correspond respectively - (16.22) cos(Aytdo—t-..4,), sin(ugt pot. py). Expansion in T is merely the expression of (16.22) as sums of terms each of the type (16.12), for the appropriate 7, s; decomposition is the reverse process of expressing (16.12) as sums (differences) of terms each of which is a cosine or sine of the form (16.22) according as s in (16.12) is even or odd. The transformations 7, /, in T apply expansion (de- composition) to a function which has been derived in I by decomposition (expansion). The correspondences (16.11)~(16.22) may be applied to ¥P by first obtaining I, 7; in I, trans- lating to P, and then reading the results in accordance with §§ 13, 14. It is shorter and easier however to omit the intermediate ¥ formulas. All steps before the last are read as in I; in the final formulas we take the PB interpretation and get specific representations of functions having parity in the form demanded by § 15. We thus retain in the working of the algorithm all the advantages of familiarity with while dispensing with all superfluous intermediate identities. From the abstract identity of %B, T it follows that the systems of linear equations next introduced have always unique solutions. THE ALGEBRA 3 OF PARITY. h3 The algorithm is reduced to the successive application of the two following. Note first that, the ¢; ¢ being positive or negative units as hitherto, Se yn €2 Lg, viele Cp Loy €1 My &y Wo, Yih Es Ws) has a unique canonical form =f (Ay, «+, Ay | pq, - - -. us), Where the sign is 4 or — according as the number of ¢;(j = 1, ---, s) which are negative units is even or odd. To express JGi+2+ -.. 4+ 4.]) in the form demanded by § 15, we start from the descomposition (not from an expansion) (16.3) 21 f (yy Bg, ooo, Bey == 20 f+ 0 Be oo +0 Be). Precisely one term on the right, that given by o = 1 (¢ = 2,..-,r), is identical with the given f. Expand this term, (8.40) Flt lot or So dpl) = f= Bel and from (16.4), by the appropriate changes of sign of the 4i(i = 2, --., 7), write down the expansions of the remaining 2" 2 terms on the right of (16.3). Reduce all terms in these to canonical form. We now have a solvable set of 21 linear equations for the 2"! functions # on the right of (16.4). The solution expresses each of these f’s as a linear homogeneous function, with coefficients +1, of the 2"! functions SO thst roid), g=410 = 2, ..., 7). Fach such expression has the same parity as the f from the right of (16.4) to which it is equated in the solution. Substituting the values thus obtained into (16.4) we have the expression of the left of (16.4) in the form demanded by § 15. Similarly, to obtain the theorem in § 15 for f(| wu, pe ++ = py) we proceed in precisely the same way from the decomposition of f(y, poy +++, pug) if s is odd, while if s is even we may start from (among others, see (10.6)), f (uy, ps, - + +, fs—1 ps). Consider now the general case of (16.11), where some or all of 4; wj(6 =1,...,#;j=1,..., 5) are conjoints. Then regarding this f as a function of 4; say f(4;/*), and of uj D4 ALGEBRAIC ARITHMETIC. say f(x uj) successively, and applying at each such step the preceding algorithm, we finally reach the actual representation required by § 15. 17. Example of § 16. To illustrate the algorithm we shall apply it to fC), L=2-+u-+» Start from the de- composition 4G u,v) = fOhHu+r)+fO+u—r) + fO—utr) + flh—p—r)), and write the typical expansion, (that of f({)) of a term on the right, JOApAtry = FO, pr) — FO uv) — Flulr, }) — flip), from which the expansions of the remaining three follow, JG = tr |) B= JF, wv D =r SG |u, 7) += J(u rv, A) =f 2, w, JO—pu+r)) = fO, u,v) + fO| u,v) — fur, )) + fri uw), Fl—p—vl) = Fh u,v) — flu») + flr, D) + Fr, The solution of the last 4 is fO, ur) = + [ fO+utr)+ fO+u—v)+fOh— utr) + fh—u—r)], 70lu, ») = 1 [—fUAutr) + f+ u—r)+ fh—utr))— fh—u—7))], Fluiy, Lb) = z [—fGFutr)+ fO+u—r)—fh—utr)+ f(h—u—-7))], Fel, wy = > [—fGtut-3)—f + u—r)) + f(h— utr) + fh—u—r))], and it is evident that the sums on the right of these have the respective relative parities implied by the notations of the functions on the left. Substituting these into the ex- pansion of f(§), we have its expression as a linear homo- geneous function of 4 functions having the respective absolute parities p ((1* 0)), p((1 1%), p((1/12)), p((1 1?) in the matric variables A, w, v of the partition {= / +p} » of {. This yeriflen S15 inthe caso o' == 3, 0 = 1, 0 = 2, DIVISIBILITY IN $$, §§ 18-21 18. Divisibility of functions in general. We saw in § 6 that multiplication of relative parities generates a semi- THE ALGEBRA 5 OF PARITY. Hd group in which there are no inverses. For the applications of PB to multiply periodic functions discussed in the next chapter it is necessary now to define a species of division for any class of functions ¢ which shall be abstractly identical, up to a certain point, with division in arithmetic. The con- sequences of the definition need be carried only so far as they relate to functions having parity, as no others enter algebraic arithmetic through periodic functions. The underlying structure in this type of division is abstractly that of class inclusion in mathematical logic or, if preferred, that of Dedekind’s definition for divisibility of ideals. Inclusion here however appears with an additional restriction which has no abstractly identical equivalent in theories of the Dedekind or Kronecker types. It will be convenient to use the simplest notations of symbolic logic, thus p Dg for p implies q, p:=:gq for pAq-qp, the dot between assertions signifying as usual the logical ‘and’; also, irrespective of the precise meaning attached to a divides 0, we shall write this as « 0, where the bar will not be confused with that which indicates parity. Let ws = (or, 90, 2) G=1,-.-.,k) be any k values of the matric variable w = (xz, y,-.-, 2) of order n. Then, if there exist % constant sealars of = 1,...,%) not all zero such that k NY S agln) = 0, == we shall say that the function ¢ og (ww) vanishes over the matriz &, of n-+1 columns and kk rows, Cr Zu oy mer 2 3 Ca La Ys Za Cn Xie Yr ccc Zk | The extension to functions of m 1 matric variables is ob vious and need not be stated. A Greek letter in square brackets, thus [§], shall denote a set of matrices, and either of 56 ALGEBRAIC ARITHMETIC. [fle = 0, |#ly (w) = 0, shall signify that ¢ vanishes over each matrix in the set [&], or as we shall say, ¢ vanishes over [§]. Each element of [§] in the above has n-1 columns, but all do not necessarily have the same number of rows, nor do any two necessarily have a common element. If ¥ — ¥ (w) is any function of the matric variable ww of order » we shall denote by [¥] the set of all matrices over which vy vanishes, and call [¢] the total set of Ww. A total set may be null. Functions ¢, ¢¥ . having the same total set are said to be equal, ¢ = ¥, and conversely, equality of functions is defined only in this sense. The set ¢ (of any kind of elements) is said to divide (or contain) the set ©, ov, if each element of » is in 6. A set therefore divides each of its elements. The sum, product of any number of sets are defined to be the logical sum, product respectively of all the sets, and equality of sets o, v is defined by GIT «CT O0:=—.0 — 7, Let 9 — gw), ¥ — W¥(w) be functions of the same matric variable w of order mn. Then the definition of divisibility for such functions is (18.1) 9 Y=: [gly = 0; that is, ¢ divides (or contains) W if and only if V vanishes over the total set of ¢. From the definitions we see at once the following, (18.2) [Ely = 0 :=: [¢]|[5], where [§] is any set over which ¢ vanishes; (18.3) ply ==: yl, (18.4) pl Wwaliy 10 ply, (18.5) YY ly I 9 = Wp, and hence, from the last two, division of functions is transitive and reflexive as in arithmetic. We shall say that ¥ in (18.1) is a multiple of ¢. Each multiple of ¢ vanishes over [g¢]. THE ALGEBRA 5 OF PARITY. 5 19. Codivisors and comultiples of functions. A func- tion ¢ which divides each of the functions vy, --., x is called a codivisor of W,..., 3; if each of the functions ¢,-..., 3 divides ¢, ¢ is called a comultiple of ¢, ..., x. Similarly for sets. In what follows the logical sum, product of the total sets [og], [W], ---, [x] of the functions ¢, ¥...-, 5 will be written [¢ + ¢ +--+], [p¢---2]. Now [gv .-- x] is the most inclusive set divisible by each of [¢], [¥], ---, [x], and [9p + @ +... 4 x] is the least inclusive set which divides each of [¢], [¥], ---, [x]. Hence each co- multiple of [¢], (¢], ---, [x] is a multiple of [py ... x], and each codivisor of [¢], [¢], ---, [x] divides also [¢ + +--+ xl]. Moreover [gt --. x], [¢ +--+ x] are the only sets having these divisibility properties and hence, in abstract identity with rational arithmetic, we call them respectively the 1..C. M. and the G.C.D. of [¢], [¢],..., [x]. We have (19.1) [Vlg] --- lx] [g] i=: [¢--- Allo]; that is, any comultiple [¢] of [¢], .--, [x] is a multiple of their L.C. M. [¢ ... x], and conversely, if [¢] is a multiple of [4-7], then [gy] iz a multiple of each of [WW], ..., [7]; (19.2) [oli -- lll ly] i=: [olf 5, which states that any ecodivisor [¢] of [¥], ---, [x] divides their G.C.D. [w+ ...4 x], and conversely, if [¢] divides [4 --.4 x], then [¢] divides each of [¢], -.., [x]. Again, fort. t fr... Jl) =... 4], by the law of absorption, and this asserts that the product of the G.C. D. and L.C. M. of the sets [¢], ..-, [x] is equal to the product [¢ --. x] of the sets. The condition that the function ¢ shall be a codivisor of YP, suis (1. 3) gl. g|x i=: [W]|lg]---[x]|ly]; and that ¢ shall be a comultiple of v¥, .-., x, (19.4) wy zle i=: lol |[¥1-- - lol ll; Hs ALGEBRAIC ARITHMETIC. and hence, from (19.1)-(19.4), (195) oly---glg i=: [¥..-4l|ly], (19.6) Plo... qle == lw... that is, if the function ¢ divides each of the functions @, ---, y, then the L.C. M. of the total sets of ¥, ---, 3 divides the total set of ¢, and conversely; also, if the function ¢ is a multiple of each of the functions , ---, y, then the total set. of 9 divides the G. C. D. of the total sets of WW, ---, x, and conversely. From (19.5), (18.3), (18.4) it now follows that (19.7) {9 Woop gf 210 — 0) 1 yr ¢ 0. that is, ¢f each of W, ---, x vanishes over the total set of , then so also does any function 6 which vanishes over the total set [WY ---x] common to the total sets of W, ---, 3. An example of 9 such that [v...3]0 — 0 is ¢ any linear homogeneous function of wv, ..., 5. The algebraic product of ¥, -.., y does not in general vanish over [¢ ... y]. From (19.5), (19,6) we now define the G. C.D. and L. C. M. of the functions ¢, -.., y. It is well to emphasize that these are sets, not functions. If in (19.5) we take [v] = l¢..- al, then [ple = 0 :=:[y..., q]lp = 0, and ¢ is now any function vanishing over the L.C.M. of [¢], ---, [x]. Since this choice gives the most inclusive [9] for which (19.5) holds we call ¢ «a greatest codivisor of vex, 0 when [gp] = [W--- 3]. Similarly, if in (19.6) we take [¢] = [+--+ x] we get the least [¢] there possible, and any ¢ which vanishes over [¢/ }-.. + y] is called «a least comultiple of ¥, ..., y. Hence we take as the unique G. C.D. L.C. M. of a set of functions those sefs whose elements are all the functions vanishing respectively over the L. C. M. and the G. C.D. of the total sets of the given functions. THE ALGEBRA 5 OF PARITY. H9 By inverting the role of inclusion in the definition of di- visibility for sets an alternative theory of division for functions, abstractly identical with the above, may be de- veloped. In this, owing to the reciprocity between logical addition and multiplication, the parts played by addition and multiplication are inverted. 20. Application of division to P. When applied to B the foregoing considerations have far reaching conse- quences, and it was to attain these, especially in the appli- cations of Pp to the algebraic arithmetic of periodic functions, that the divisibility of functions was devised. Before pro- ceeding it will be well to notice the remarks at the end of 3 6. Let 4 — w+v+4...4 0 be any partition of the matric variable 4. Then any function F(u, », ---, ¢) (not necessa- rily possessing parity) is, with respect to simultaneous change of sign of all its matric variables w, », ---, o, a function of 4, for the change of 2 into —24 induces the change of F(u, Hii sositiy 0) into Bn, Sarin ~~}, Suppose now that Ff = 70), g g(l 4), 4 as above, are entirely arbitrary beyond their indicated parities p (2), p( 2), and let # Fly, v, +++, 0), Gu, v, -.-, g) be any whatever functions having the respective parities p (2), p( 2) as just explained. Then (20.1) zn gle for, 7, g being arbitrary to the extent stated, AIF = 0.1717 = 0, [gly = 0.00 = 0, Thus, in our technical sense, F, ¢ are multiples of f, ¢ respectively. Again, it is clear from (13.1), (13.2) that each term in the expansions there has the same parity p(Z]) or p(| 2), where 4 — A; +234... +44, as the functions expanded. Hence if f, g are as in (20.1) and F, G now denote any Junctions having any of the parities chosen as just indicated 60 ALGEBRAIC ARITHMETIC. Jrom the expansions of f(A), f( 4) respectively, it follows that (20.1) holds for these F, G ond therefore if f= 0, 9g = 0 over any set of matrices we infer immediately that F = 0, G = 0 over the same matrices. The relations [fF = 0, [91G = 0 are thus instances of [f1f = 0, [glg = 0 respectively. According to our definitions F', ¢ are multiples of f, g respectively. To illustrate the theorem we write out for a few simple functions their multiples obtained by expansion. mm Function Multiples pv S42) Se, vl), fw, v); p+ JS(2) S|), fm); otra Ff) So, v, 0), fle), 0), Su, 0), flo|n, v); ptv+to F125) Sle, u|v), f(v, lo), Sw, v, 0), fv, 0|0), in all of which the f’s are entirely arbitrary (and hence not connected by any relation) beyond the parities implied by their notations. Illustrating the last for example, we see that 26 Fmitvit eo) = 0 implies each of the following, > cif (we, vi, 0) = 0, 2. ally, ald = 0, ifn mv) =0, X.6fe, mle) =o. Note that conversely the last four together, but no fewer of them, imply the theorem from which they were inferred. For, the /’s being arbitrary to the extent stated, we may choose for them the approximate fs, having the same respective parities, as those occurring in the expansion of f(A + p+») according to § 15. This remark is now generalized. Let A = w+v+ ... +o, and let Ji (we, Vee; 0) (J =1, 2, k) denote all the fs occurring in the expansion of a definite one f(A) of f(A), f(A). Then THE ALGEBRA 3B OF PARITY. 61 (20.2) alive) =0 G=1,2,..,0) together imply 25; cf (4) = 0, where 4, — uw, +v,+... +o, and no set of fewer than k of the relations (20.2) imply this conclusion. In all such theorems the complete arbitrariness of the functions except for their definite parities is to be noticed, as it is from this that their power and utility springs. We can now state a general theorem. Regarding f= f(Q, u, -- ce, v|Q, 0, «+, 1) as a function of each of the matric variables by +o Tam turn, partitioning these in any way, and applying the theorem of § 15, we can write down from the vanishing of Jf over any matrix a system of equations for other fs, arbitrary in the matric variables of the partitions. of relative parities in the partition variables given by the expansion of the original f; conversely, from all the latter equations, but Jrom no fewer of them, we can infer the original. To ustrate; let 4 = 4, 14s, 0» = un, Then we have the following multiples of Sp): Jlhy, Alin), FU 2y, 43, 0), fa, Hols), FUL, wali), SPs 29, py ue), Sl | Ay, Ag, py), Say 2g, p13 | );, Sls) A, 2, 1), giving in all 9 multiples of f(Z w), itself included, in which each of A, w is partitioned into not more than 2 conjoints. The process can be continued so long as at least one iy My is of order >1, and at any stage there are the possibilities of partitions of 4; into /; conjoints, where I; <1; I; — the order of 4; and similarly for wu, Finally, observing that any linear homogeneous function of any multiples of f(4 | ) is a multiple of f(2 w), we get further multiples, and the like applies to the general case. The complete generality of the multiples cannot be too strongly emphasized; the above theorems include all possible inferences concerning the vanishing over a given matrix to the vanishing of other 62 ALGEBRAIC ARITHMETIC. functions over the same matrix. It is necessary only tha the parity of the functions be the same. ; The general theorem can be given a slightly more general appearance by observing that in the multiples of a given JQ ) or S( 4), where A= pv ..-to, the oy Ws vr, 0 may be replaced by au, bv, -.., co, where a, b, ---, care scalars different from zero. This however is a special case of the theorem, and it illustrates the above remarks on the generality of the functions involved. By successive applications of expansion and decomposition to the arbitrary fs of parity p(ai, «++, ar| by, -- +, bs) in any sum of such f’s vanishing over a given matrix we can generate a closed set of vanishing sums for arbitrary functions having parity. The number of such sums is an ordinary partition function of the a; b; which depends upon the combinational function 77 (w) mentioned in § 26 of the preceding chapter. 21. Parity transformations. A transformation of the matric variables of a function f having parity which leaves invariant the absolute parity of fis called parity transformation; and the result of the transformation a parity transform of the original function. These play an important part in the deduction of special cases of the general theorems inferred from periodic functions in the next chapter. There seems to be no simple means of defining the most general parity transformation, but we note the following useful instances. Yet 2 =— (%,.-.-, z,) be a matric variable of order », and let (2D, 52) (7 = 1, ---, nw) be arbitrary of the indicated parities. Define new matric variables Z,, 7; of order n, by Zo (f(z ); Jo (2 honey Jalz » Z (g:( 2 nl, -, 2 2). Then each of f(%4 |), f(Z), F(Z |) is a parity transform of f(z|), and F(Z) is a parity transform of f(z). More generally, if the respective orders of the matric variables ‘nb a, hil =1,.-.,7; 5 =1,..., 9), a parity trans- form of f(A, ---, Ar | py, ---, ps) is given by replacing each THE ALGEBRA § OF PARITY. 63 variable in 4; by an arbitrary function of parity p(a; 0) or p(0 a) in all the variables of 4; and each variable in gw; by an arbitrary function of parity p(0 0;) in all the variables of 1; (i =m eng J w= Los, 5). From § 20 it is clear, that if F is a parity transform of the arbitrary f having parity, then fF, ALGEBRAIC PARITY, GENERALIZATION OF %, § 22 22. Parity in an algebraic number field. The s0- called functions with recurring derivatives of Olivier, Nicodemi, (Glaisher, Appell and others suggest a wide generalization of the concept of parity as defined in § 2. The functions just mentioned are the simplest instances of certain functions in n variables with » periods related to an algebraic number field of degree n. The functions of Olivier and others refer to the very special case in which the field is generated by a primitive nth root of unity. In the general case the field is defined by any irreducible equation of degree un. There are associated with the field sets of » functions which are linearly dependent when the variables are multiplied by numbers in the field. The theory of parity as developed above refers to the simplest of all instances, namely that in which the equation defining the field is x-+1 = 0. As the generalization is extremely extensive, and as it leads to a new domain of algebra, we shall not pursue it further here, but may refer (for certain algebraic details) to a paper in the Quarterly Journal (1926-7) on N-fold periodic functions connected with an algebraic number field of degree N. CHAPTER 111 THE ALGEBRAIC ARITHMETIC OF MULTIPLY PERIODIC FUNCTIONS THE PRINCIPLE OF PARAPHRASE, §§ 1-5 1. Application of parity to periodic functions. The algebra Pp has immediate applications to the algebraic arith- metic of n-fold periodic functions, » = 1. When n = 1 the application leads to the numbers, polynomials and functions of Bernoulli, Euler, Genocchi and Lucas in a rather un- expected way, also to the novel generalizations of all these implicit in the algebra of elliptic functions. This application is developed by means of BB and a subalgebra of the latter abstractly identical with 2. But as the application of to the n-fold periodic functions when » 1 leads to a far richer theory than that just indicated, we shall discuss the case n >>1 alone. Under n-fold periodic functions we include the pseudo-periodic, for example the doubly periodic functions of the rth kind, » > 1, of Hermite and others, and the theta functions of p >>1 arguments. Unless otherwise stated theta shall mean elliptic theta function, and theta quotient any rational function of theta functions. Hence in particular the elliptic functions and certain of the doubly periodic functions of the rth kind are theta quotients as here defined. The n-fold periodic functions are connected with algebraic arithmetic through the circular functions in the following manner. Consider first the case » = 2. In the usual notation any theta quotient has a twofold type of expansion, first as a power series in ¢, the coefficient of the general power of g being a function of the exponent or of its divisors and circular functions of the arguments, second, as a Fourier series. The latter, upon expansion of the coefficients of the trigonometric terms into power series in ¢ yields a series of 64 MULTIPLY PERIODIC FUNCTIONS. 65 the first species, which we shall call an arithmetical expansion. In the above we have neglected temporarily terms involving reciprocals of sines or cosines; see § 13. Such expansions are not current in the literature because their principal interest is for the theory of numbers, especially for that branch of it in which we are at present interested. The arithmetical expansion of a theta quotient is unique (including the terms in reciprocals of sines or cosines). If in any identity between theta quotients the several terms be replaced by their arith- metical expansions, and if in the result coefficients of like general powers of q be compared, we get a trigonometric identity which, as will be shown, implies and is implied by an identity between arbitrary functions having parity. The identity between parity functions is therefore formally equivalent to the given theta identity; it presents all of the arithmetical information implicit in the theta identity in a concise, sug- gestive form, and conversely, from the parity identity that between theta quotients can be recovered by specialization of the parity functions to circular functions having the same respective absolute parities. When n = 1 there are no arithmetical expansions as above defined. Hence the algebraic arithmetic of the singly periodic functions is radically different from that given by the case n>, When »n => 2 there enter, in place of the single parameter ¢ several, say » >> 1. With respect to these r parameters it can be shown that arithmetical expansions exist and that, proceeding from these in the same way as when n — 2, we should reach systems of not less than » arithmetical identities between arbitrary functions formally equivalent to a single identity between n-fold periodic functions. The r-fold gene- ralization of € mentioned in the first chapter appears naturally in this connection. But when n >> 2, except in the case where no theta function occurs in the denominator, the arithmetical expansions have not been obtained, and it seems to be a matter of considerable difficulty to elicit from Appell’s expansions (n = 4) forms appropriate for arithmetic. Were these 5 66 ALGEBRAIC ARITHMETIC. expansions completely known we could more than double the extant arithmetic of systems of quadratic and certain higher forms in any number of indeterminates at one step. 2. Parity functions. Thus far we have discussed functions having parity without distinguishing the instances according to those, A;, of A, in which the matric variables lie. The expansion and decomposition theorems in require only that the matric variables be in a ring R of A. Let R; be a ring in the instance 2; of A, and let / be a function having parity. Then, if for each set of values in RR; of all its matric variables J takes a single definite value in 2;, and if further f vanishes with each matric variable with respect to which f has an odd absolute parity, we shall call fa parity function in A; or, when 2; is understood, a parity function. In general A; will be ¥,, although a considerable part of the main theorem is proved in A. It is emphasized that beyond the three re- strictions stated, namely that fis uniform with respect to values of the matric variables in the ring concerned, f possesses parity, J vanishes with each matric variable occurring to the right of the bar in its symbol f(* | 7), the parity function fis completely arbitrary. 1f the parity of f° and its uniformity with respect to integral arguments be preserved, but further conditions be imposed upon f giving, say, #., we shall call F a restricted parity function. 3. Principle of paraphrase, first form. Let a; b; (¢6=1,...,7;5=1,..., 5) denote J integers >>0, and write att =m, ht... tbs =m, yr =46, s = 0, 0 = ote, 0 = d +9, as in chapter II § 4. In what follows the x, y with suffixes in the J matric variables & n with suffixes are o independent variables in §., = la, a, 2) n= a; ==] .,. 7} BE np) bE=b; j=1,..-, 0 the u,v with suffixes in the following nd matric variables with suffixes are no rational integers, MULTIPLY PERIODIC FUNCTIONS. a7 ap = (war, Wisk, ++, wo) (0 = ag; 4 1, 20,7), Bj mr (jit Vjokes °°» Uji) (0 mm bjs J = 1, eu, £), F=1 7-7) : In writing functions ¢ ((¢&)) of scalar products («&) we shall omit one ( ) and write simply ¢ («&). From the above matric variables form scalar products as follows, {ow 5) == wy, Zi + ior io + + + + + tia Lia Bix 1) = vi y+ vor vio + - - + vox yj, aw, b= i=1,-..,7; j=l... a1 Le=1...%) ; bj J Let each of the following f.¢, 2 be a parity function in &,, the ring R, in which the matric variables lie being that of the rational integers, (3.01) f(&, En, 90), gr, &l)y Rg, 90), of which the respective absolute parities are (302) plm,---, wp|ly, ++, bs), plas, +, 00), p@OBy,---, Bs), and the orders, degrees are respectively B07 we. 8 wy 0. Let gu, Ix, tx (k = 1, .-., un) be constant integers. The generalization of what follows to the case where the gx, I, #: are considered as elements of $F. is not essential, as it can always be reduced to the simultaneous assertion of one or more theorems of the type stated. We can now formulate the principle of paraphrase. If a particular one of (3.11) 2h 11 cos (eer; &) 1 sin (8x al == 4), i—=1 J=1 (3.21) ; S11 cos (ay; & a) = (), k=] =1 3.31) S11 sin (Bj nj, 3 — 0, kt=1 j=1 68 ALGEBRAIC ARITHMETIC. is an identity im all of the x, y variables (with suffires) occurring in the &, nu; present, then that one implies and 7s implied by the corresponding one of 9% (3.12) = qr f (typ Uses al Cpl Bis Bar, ality Bic) m= 0, T=1 n (3.22) - lig (ear, Caky = + +5 OL) =, b=1 n 3.32) > til ( Bik, Bary = +, Bic) = 0, i=—1 where f, gy, h are as in (3.01). : The implication of (3.51) by 3.72 (; = 1, 2, 3) is evident since the trigonometric products are instances of f, g, & re- spectively. Again, (3.12), and hence (3.11) implies (3.32) but not (3.22). Hence it remains only to prove that (3.11) implies (3.12) and that (3.21) implies (3.22). The last two proofs will presently be reduced to that of the special case of (3.22) in which ¢ is of parity p(n | 0). In the above statement the «, 8 matrices are in &,. We shall prove much more than this. It will be shown that if the a, 8 matrices are in A, and the circular functions also are in A, then (3.21) implies (3.22), and when s = &, in (3.11), (3.31) is even, these imply (3.12), (3.32) respectively in A. The two excepted cases of this generalization from F, to 2A when J; is an odd integer also appear for several reasons to be true, but I am unable to prove them in 2. The extensions to 2 are however only of algebraic interest, as they can never occur in ordinary analysis or in any of its arithmetical consequences, for such analysis is based on either §. or %,. In particular the algebraic extensions to A are irrelevant for the arithmetic of periodic functions. Simple applications of the principle are given in §§ 14-16. 4. Second form of the principle. Before proceeding to the necessary proofs of § 3 we shall state and prove a modified form of the principle. This modification will cover most, but by no means all, of the applications of the parity identities that will ordinarily suggest themselves. - What MULTIPLY PERIODIC FUNCTIONS. 69 follows differs from the first statement in this important respect: it us further postulated that in some domain the parity functions have convergent power series expansions. The modified principle applies therefore only to restricted parity Junctions (analytic), the principle in § 3 to unrestricted (not necessarily analytic) parity functions. For what follows 1 am indebted to Professor G. Y. Rainich; it has not been published elsewhere. let F F(x, ---, x) be a function of » independent variables x, .-., x, in &. or &- which has a power series expansion ; \Y . . i in (4.1) Zr a ~ AG, ee, in) x ss In converging in some domain A. In general some of the coeffliciels AG, ---, 4) lp=0,1,--15=1, -.., n) will be zero. Let 6 = Gn, +, 2) also have an expansion 9) x = a Tn 4.2) Bly 00,9,) Ll convergent in 4. The coefficients Ade, 0), B == Bl]. 7) of two terms in (4.1), (4.2) are said 1p correspon! when and only when GG, ---,4) = (1, 100) (matric equality), and G' is said not to exceed the type of F when those coefficients 5 which correspond to zero coefficients A also vanish. For example, no even analytic function exceeds the type of cos, since all those coefficients of a power series representing ~ an even function which correspond to the vanishing coefficients (those of x, 2% x° ...) in the expansion of cosz must vanish. Again, #® does not exceed the type of cosz, but not con- versely since the coefficient of 2* in 2* vanishes while the corresponding coefficient in cosx is 1/24. Suppose now that for a given F(x, ---, x,) there exist (n+ 1)m constants Grrr ty Cm; Us ~>vs tim : (j = 1, ..., n) such that (4.3) 2 F(a Zi, ++, Ay Zn) = 0 70 ALGEBRAIC ARITHMETIC. is an identity in the 2; ( = 1, ..., »). Then, if a function F satisfies an identity (4.3) the same identity holds for every Junction G which does not exceed the type of F. To prove this, replace F(z, ---, z,) by its power series. The left of (4.3) takes the form of an identically vanishing power series. Hence each coefficient vanishes. But these coefficients are linear combinations of terms of the form (4.4) AG, 0, 4) wl. a2, namely, D2 uAl, hry 9) ao: . ar, (4.5) ye YL > : = Al, AP os Ss Since the last product vanishes, at least one of A(, ---, 7), 2c at --- as must vanish for every set of values of (#1, +++, 7,). Consider now the expression corresponding to the left of (4.3) which is derived from a function G (xy, ---, 5) which does not exceed the type of /. On substituting for its power series expansion we get a power series whose coefficients are of the form nn (4.6) BG, 1) =~ ¢ a“ ns a Compare (4.5), (4.6), and recall that (4.5) vanishes in all cases. It for a particular value of (5, ---, 4,), (4.5). vanishes on account of the > factor vanishing, then (4.6) vanishes because its > factor is identical with that of (4.5). If on the other hand (4.5) vanishes with 4 (7, ---. 4,) for the particular value of (4, - ++, 4n), then B(4, ---, %,) vanishes for the same value, since does not exceed the type of F. Hence in all cases (4.6) vanishes, and therefore mn (4.7) > Gj G (at) L1s +c, Unj Zp) = 0, J=1 which is the theorem. MULTIPLY PERIODIC FUNCTIONS. 71 To apply this to the restricted principle, consider an analytic J (= possessing a convergent power series expansion in all its variables) having parity p(2|3). This will suffice, as the proof for / having any absolute parity is precisely similar; we start in any instance from the simplest circular function having the same parity. Consider then 4.8) fx, y;u, v,w) = cos (x+y) sin(u-+v+w), where x, y, u, v, w are independent variables in #,. The sum of the exponents of = and y in each term of the expansion of (4.8) is even, the like sum for u, v, w is odd; the co- ig of any term violating these conditions is zero. Let =z = f(z, »)|(u, v, w)) be a restricted (as to analyticity) parity EH having, as indicated, the same parity p (2/3) as (4.8), which is defined for integral values of its arguments. We may now consider an analytic function which will assume the same values as f° for integral values of the arguments.* The latter function will not exceed the type of (4.8), and hence it will satisfy any aes of the form (4.3) which is satisfied by (4.8). The following illustrations (due to Professor Rainich) throw further light on the nature of the restricted principle. An example of (4.3) with n=1, m=2 is f(x)-}f in which the coefficients c, nave co =a = 1, n = —gp=1, This is satisfied by every function which does not exceed the type of sin, namely, by every odd function. Consider next Fat) —F@) — f+ 10) = 0 Here F(r,y) = f(x + y), n = 2, m = 4, and the coefficients C, 4, Are Ci. — — Co EET Cn T= Cy == 1 5 hh == t= 1, a =a, — 0, by — Dy 1, bs = by = Q. * Appeal is made here to a well known theorem of Borel (proved by him, however, only for functions of one variable). 72 ALGEBRAIC ARITHMETIC. The identity is satisfied by any function which does not exceed the type of 1+4z—+y, for example A(z+y)+ B, where 4, B are independent of x, 4. In the following example n = 1, m =— 3. Take Flax) fa) flea) | « bh gr l= 0, | 1 1 1 | as the instance of (4.3). Here 6 = h—e, Go = ¢—a, 6G = a—2>0, w=, w= 0, wily =, and the functions f are linear, 4 -+ Bx, or functions which do not exceed the type of 14 x. As a last example, consider functions illustrating the al- gebraic parity mentioned in § 22 of Chapter II. Let « be a complex cube root of unity, and write Ll) = expa-tod expen? expels (j=0,1,9). Then in the development of fj(x) only exponents = j mod 3 appear. Consider now, for example, functions which do not exceed the type of hlety flvto Aten filo). Their expressions will be characterized by the vanishing of certain coefficients, or by the property that if z, y are re- placed by «x, ey, the function is unchanged in value, while if p.q,r are replaced by «p, «gq, «r the function acquires the factor «, ete. 5. Algebraic lemmas. We return now to the principle as stated in § 3 and outline a purely algebraic proof. The following differs from the original proof given in the 7rans- actions, vol. 22 (1921) p. 1, and it applies to more general situations. The algebraic lemmas themselves are not without interest. MULTIPLY PERIODIC FUNCTIONS. 3 A reference to any treatise on algebraic equations, or to MacMahon's Combinatorial Analysis, will show that Waring's (or Girard’s) theorems on symmetric functions are mere identities in A, to prove which are required only the postulates of an abstract field and their immediate con- sequences; this is also evident from Kronecker’s theory (see for example Konig, Algebraische Griffen) of elimination. We shall therefore state and prove our lemmas for 2, although the applications which we have in view refer to J in JF. or &r. let ¢; G=1,-:.,0), y;{5=1,---, 8) be sets of elements in A, and write Ed -Lalt onbl, mn pl yi) rhachis {m= 0,1,:..), (—1)* pi, = the kth elementary symmetric function (k=—1,...,0) of the ¢; ¢=1,..., 0), and (—1) 1; = the Ith elementary symmetric function. ((==1,..., 8) of the (G=1, --., 8); Po = my = 1 = the unity in A. Without loss of generality assume 8 = hh. We shall prove that (5.1) Sm — Op (m =1,.. ois s) implies that 8—0b of the y; (j= 1, -.-, 8) vanish, and that the remaining b of the y; are a permutation of the ¢; (i =1,..., b). Let the x; be in A. With the usual convention 0! = 1, and with >’ referring to all sets of m integers »; (t=1, .-., m) each =O such that », + 2r + 3r-+ -.- mem = m, write P (71, Low: > vx, Zn) = > [en Xoiitd nn Bn) 5 fini) | AL Se ei ] 7 7 Ia = ( 1 2 in 2 =x Tey 2.0, phim Hn 1 ) 2 \ ne Xe 2s Lm PE ” r, ” Aon, — ri! 12 > Pd; Bo = 1 L 2 A w. In the instance J. of A, ¢ is a familiar expression occurring in connection with Waring’s theorems. Write 9p (51, So moi, Sm) me Pos 9 (01, Oi, Ton) =U). 74 ALGEBRAIC ARITHMETIC. Then, by (5.1), we have in particular (5.2) P= 4, r=1,..., 8. Again, by a well known theorem on symmetric functions in &e which, as pointed out, goes over unchanged into 2, 5.8) B=yp(i==1,.-. 0), L=000, We=nl==1..0) Hence the following is an identity in the (Kronecker) in- determinate uw, B bh p—0 = : aL: v a; (5.4) 2 70; wh = piwP= + Popul ai J== J=0 the second sum on the right being omitted when g — b, as indicated by the limits. By (5.3), (5.4) is equivalent to (0 —1r1) (uw which proves the theorem. (In this proof we have not assumed the fundamental theorem of algebra in its Se form, but have used the abstractly identical consequences in A of this theorem, necessary for the above inference, in the form given by Kronecker. An accessible introductory account of the relevant algebra is given in Wedderburn’s paper on Fields in the Annals of Mathematics 1922.) Let next 2 =— (z, ---, z,) be a matric variable of order in Ws leitheon, spli=1,.- by j=1... 8; b= Yio) be scalars, and for m = 0, 1, ... write ee (0—y8) = w= (we) (0 —3)-- (—11) 2 /2 Ci {ens tomy - =>, 0s), L (700 ¥10s +2 =; Vids Sm (CLL) 5, = (Ts yrl.. (] sy where (wz)™ is the mth power of the scalar product (wz). We shall prove that (5.5) S. = —~ (m — 3. Se iy 3) implies that 8—0b of the I; are each equal to (0), (the zero matrix of order n in A), and that the remaining b of them are a permutation of the C;(i = 1, .-., b). MULTIPLY PERIODIC FUNCTIONS. 5 For, by the lemma just proved it follows from (5.5) that p—0 of the (32) ( = 1,--:, B) vanish and that the re- maining b are a permutation of (G2) (= 1, ---,'0). . But a particular (Iz), say (Iz), since z is a matric variable, vanishes only with 7, that is, only when I. = (0),,. Further, a particular pair of the (Iz), (C;z) are equal, say (Jz) = (C,2), only when 73, = C,, for the same reason; which is the theorem. Observe now that Sh, is the sum of the mth powers of the squares of the (Cz) 6 = 1,..., 0), and similarly for 2m. Hence, by the last lemma, if . (5.6) Som = Som (m = 1, -. 8) we reach a conclusion regarding the (C;2)?, (L322 (i =1, ..., b; J = 1,..., 8) precisely similar to that which follows from (5.5) for the (C;z), (I;2). Since z is a mairic variable we have then the chain of implications, pe) = ( Cy?) SN: T, = CO; (Ip2)* === {C, 2) S570 I (L,2) = + (C,2), =: 1 == LA, 2D: fp) = fC), where f(z|) is a parity function, of parity p(n|0) in the scalar variables of z, or p((1|0)) in 2. Without loss of generality let the #— 0 vanishing 7°. be Ii ({=1,...,8#—05). Then, from what precedes, it follows that (5.6) implies bh g—0b f GT) 2G) =Z 50+ 2 FT). The last has an immediate and useful extension. let the ¢, be integers, and write Le loltiel 4 tla Then, by suitable transpositions, (5.8) > pC 2" = m= 1,-2.7) p=1 76 ALGEBRAIC ARITHMETIC. can be reduced to a relation of the form (5.6), to which we can apply the lemma proved for (5.7). By retranspositions in an obvious way to a form corresponding to (5.8) the final result of this application of (5.7) shows that (5.8) implies t (5.9) DS afl) =0, p=1 Let the c, be as above, and let ¢ be an integer. If (5.10) Xi = we may consider, in the following proof, ¢ = 0; for if ¢c< it suffices to change signs throughout (5.10). Applying the results already proved we shall obtain the first principal lemma: (5.8) and (5.10) together imply ; (5.11) 2 fC) = FO). The coefficients ¢,, ¢, by the above remark, are any rational integers. By an obvious extension these coefficients may be any numbers in §,, for the latter case is reduced to the former on first clearing of fractions in (5.8), (5.10), and finally, in the statement deduced from these by the first principal lemma, corresponding to (5.11), dividing throughout by the common denominator used in the first step. It will be sufficient to outline the steps by which this lemma is reduced to the preceding. By transpositions, if necessary, and by an application of the remark on (5.10), also by writing ¢; = 1+1+... +1 (p times), or the negative of this if ¢, <0, the hypothesis is reduced to the form which implies (5.7). By reversing the above trans- positions and resolutions of coefficients into sums of units, we recover from the equality corresponding to (5.7) the required (5.11). The first principal lemma is implied by the eveness of the exponents 2m (m — 1,...,k) in (5.8), and it may be anti- cipated that MULTIPLY PERIODIC’ FUNCTIONS. 7 t 8.12) - Cp {Cs Zyl Ee 0 (m EE 1. olny k) p=1 implies (corresponding to (5.11)) t (5.13) 2 alll) = 0, where the parity function f(|z) has parity p (0%) (or p((01)) in 2), but I have not proved this in 2. TI shall therefore circumvent its use in proving the principle of paraphrase in ¥, by making all instances of which (5. = is the type depend upon (5.11). On account of its interest we conclude these lemmas with a purely algebraic theorem, due to Professor C. F. Gummer, which can be taken as a basis for proving the first principal lemma in F. or F,. It was shown by Laguerre that the real equation n n, m2 tari... = 0, in which the n; are rational and 7, > mn, > -.., has not more roots >> 1 than the number of variations of sign in the sequence a, a+ az, a+ a; as, and the proof of this theorem can be extended to irrational 7;. Let x = ¢5, nj = log. ¢, ¢;>0. Then it follows that ) E 7 ac = 0 has not more positive roots in & than the number of variations in the sequence. Hence, if the J;, ¢; are positive, 1] >" 7 PS is not true for more than p—1 positive integers & unless it is an identity. 6. Application of the circular functions in (. It is easily seen that Jf can be obtained by operations in B or € 8 ALGEBRAIC ARITHMETIC. from the circular functions in 2 in a manner abstractly identical with that which is sometimes used to derive T from the power series expansions of the sine and cosine in J, or J. If 2 is in A, we define sinz, cosxz, by 20 2n+1 cosy = 35 (— Wis sine = 25 (—1p rT and the operations by which such functions are combined to yield ¥ in A are in either B or €, preferably the former, see chapter I § 22. If A is replaced by either of its in- stances 3; (j = ¢, r), these functions and all operations upon them are as in common analysis; in 2( they have no numerical significance. We recall that (zw) is the scalar product of the matric variables z, w, and that ¢ ((zw)) is written ¢ (zw). With the notation of (5.11) for Cy, z we have the following. If t (6.1) 2 0 eos, 2) == ¢ p= is an identity in z, so that (6.1) holds for all values of the inde- pendent variables zj(j = 1, ---, m) in z, then (6.1) implies (5.11). For, if in (6.1) we replace z by tz, where ¢ is a scalar parameter, and equate coefficients of like powers of #, we recover (5.8), (5.10), which together imply (5.11). Note that by equating coefficients of #”(m = 0, 1, --.) we get in addition to the relations required for the proof an infinity more which are unnecessary. The superfluous relations are implied by the necessary, as may be shown by the lemma just proved or independently. Thus the cosine identity (6.1) is sufficient but not necessary for the proof of (5.11), and ° this appears to be the algebraic equivalent of the analytical concept of type excess for functions in § 4. Now by decomposition in T each of (3.21) and (3.11), (3.31), provided in the last two that s be even, can be re- duced to the form (6.1), from which follows the correspon- ding (5.11). From the last, by the identical transformation 3 in P abstractly identical with the expansion formula in MULTIPLY PERIODIC FUNCTIONS. 79 which in each restores the decompositions in I to the functions decomposed, we infer the principle of paraphrase in 8 8 for (3.22) in oll cases, and for (3.12), (3.32) in those cases when s — 0; is even, in the extended form in which the a, 8 matrices in A. There remains then only (3.32) with s odd, since evidently (3.12) with s even is reducible by the identical algorithm in PL to dependence on (3.32). At this point we pass from A to its instance Fy. The proof for , is easily completed by applying the identi- cal algorithm in PB to the following remarks. From the definition (§ 2) of parity functions, such a function f(| zy, ---, za), of parity p(0]17) in the variables (5 = 1, :-., wm), Is equivalent to (z; 2; « - - 2a) f(21, 22, - - +, Za |), Where f(z, « « +, 2a)) is a parity function, of parity p(1]0). Consider now the instance of (3.31), (3.32) in which the absolute parity of 7, indicated in (3.02), is the special one p(0] 1%). Each matric variable in 7 is now of order 1. By partial differentiations with respect to each of the s independent variables now in (3.31) in turn the I identity (3.31) is reduced to a cosine form to which the preceding result can be applied immediately. Each term in the paraphrase is of the form cB By Ta? Bs f(By, Bs, Sete Bs D, mm ef ( Bi, Ba, ge Bs), by the first remark. The above algebra is legitimate if and only if the 8;(j = 1, -.., s) are rational integers as assumed. Hence the paraphrase of (3.31) into (3.32) is proved in the special case when % in (3.32) is of parity p(0| 15). Thence, by the identical algorithm in ®B, the general (3.22) is established on observing that in any ¥ identity whose right hand member is zero, obtained by expansion, the set of all terms having a given parity vanishes independently of the remaining terms. Hence we have proved the principle in § 3 in all cases, and we have extended it from §, to 2 for parity functions whose odd degress 0, (= s in the notation of § 2) are even integers => 0. As already remarked it seems highly probable that the extension to 20 holds also when J; is odd, but this has not been proved. 0 ALGEBRAIC ARITHMETIC. EXTENSION OF THE PRINCIPLE TO HIGHER FORMS, §§ 6-11 6.1. Odd and even algebraic forms. We shall say that the matric variable z = (2, ---, z,) of order r hi an integral value when and only when the z(j = 1, ---, 7) are rational integers. Thus the ag, 81 (G = : LE d=... 5 L==1, +. un) in (312-352) are integral values of the matric variables &;, 7; in (3.01). The notation w, wy, vw, &, uj, +++ in what follows is as in § 3. A rational integral algebraic function, not necessarily homo- geneous, in / indeterminates with rational integral coefficients will be called a form of order k. Forms of preassigned orders and parities can easily be constructed. For example, the indeterminates being the x, y with suffixes occurring in the &, 7; we get from (3.11) the form (6.1) 3s qx 1 (eens $i) 11 (Bj) i=1 J=1 having the order w and the parity play, ---. a, by. ---. bs); from (3.31), (6.2) pA i=1 a form of order w, and parity p(y, ---, a); from (3.31), (6.3) allure, =1 a form of order », and parity p( by, ---, bs), in all of which u;, yli=1,---,7;4 =1, .-., 5) are arbitrary integers > 0, the (an &;), (By; are scalar products as always, and in (6.1), (6.3) Bi is different from the zero matrix of order bi(j = 1,...,5) in F By expansion and decomposition in 3 we can write down from (3.11)-(3.31) any number of forms having assigned orders, degrees (d, dy, d;) and parities. Let A be any form of order w whose indeterminates are the x, y with suffixes occurring in the matric variables &, MULTIPLY PERIODIC FUNCTIONS. 81 mli=1,-»»,r;5=1,.--,38), Then if in the &, 7; this - form has the first of the parities (3.02), we shall write 4 = 405, ois sl, 5 | 7, voy 7s), which is said to be even in each of §(G = 1,..., 7), odd in each of (fj = 1, ..., 5). Similarly for the forms B, C, n= BE... |), {= Clq, 7s), of the respective orders wy, w;, even in each of the &; odd in each of the #; respectively. Since A, B, C are instances of parity functions the algebra 93 applies to them. In par- ticular an arbitrary form can always be represented as a linear homogeneous function of forms having parity. In general a form has no parity; those having odd parities, p( | by, - - -, bs) are of special importance in the applications of the principle of paraphrase. 7. Compound representation. Tet 2’ be an integral value of the matric variable z = (, ---, 2.) of order r. Then, without loss of generality (see chapter I, § 9), any form A(z, zs, ---, 2») in the r indeterminates z;(j =1, ..., r) may be considered as a function of z, and it may be written 4 (z). I 2 is such that A(z) = 7, the integer = is said to be represented in A(z) by 2’; if no integral value z” of z exists such that A(z”) = mn, then = is said to be mot represented in A(z). Let now A(z) be odd in z, so that we may write A(z) =— A(z), and let A[n] denote 1 or O according as » is or is not re- presented in 4(|z). Then An] = A[—n]. For if » is represented in A(|z) by 2, so that 4A(|Z’) = =, then —A(Y) = A(|—7Z) = —n gives the representation of —mn in A(|2) by —Z/. If next » is not represented in A (|z), if possible let 2’ be a representation of —n, so that 4(|2') = —n. Then, by what precedes, —z’ is a representation of nu, con- trary to hypothesis. Again, since A[n] = A[—mn], and A[x] takes a definite value for each integral value of the indeterminate x, it follows that A [x] is a parity function having the absolute 6 82 ALGEBRAIC ARITHMETIC. parity p(1|0), and we may write Ax] = Alx|]. Hence the function Alx|] of the indeterminate x, which takes only the values 0,1 for integral values of x, and which is such that Aln|] = 0, 1 according as the integer m is or is not represented in the odd form A(|z) of order r, is a parity Junction of x having the parity p(1 0). With z= (#1, --+, 2) as above, lei 4,(2) G=1,-.-, 7) denote odd forms, each having the relative parity p(|z) as indicated by the notation, of the respective orders oj (j=1,---,7), so that 4;(z) is a form in precisely w; of the zj, 02 by means of K. In all cases the application of L, K introduces forms of higher degree than the second. Proceeding in the same way with 7 and K, we restate (3.12)-(3.32) with reference to Six, in which L is applied first, and similarly for @xz. Taking now either the same or different K, L we can start from Srx, ©xr, and derive paraphrases integrated over x yd fo E> CLL, OLE, Orie, Crug, cv and so on indefinitely. In this way we escape from the traditional limitation to forms of degree + 2 of the applications of the elliptic and other periodic functions to algebraic arith- metic. It is to be noticed that the odd or even degree of any Jorm concerned is an arbitrary constant odd or even integer. The applications are so numerous, and so readily made, that we shall take space in the following for only a few of the simplest to show their nature. APPLICATION OF THE PRINCIPLE TO THETA QUOTIENTS, §§ 12-16 12. Arithmetical expansions of theta quotients. We shall indicate a few of the more prolific sources from which arithmetical expansions (see § 1) can be derived. Of the first importance in the present connection are the doubly periodic functions of the second kind, Poy (x, 1), where Fe ) = =— Ed {, q) {0 — = 0, 1,2 = 3), U1 Ya {x +9 — Papy (, 9) = 7 (wn) 3, LT = Qagy (x, Y, 9), MULTIPLY PERIODIC FUNCTIONS. {9 for the following 16 values of the triple index «gy, 001, 010, 023, 032, 190, 111° 122° 133, 203, 212, 221, 230, 302, 313, 520, 331, as by means of them we introduce in the simplest way parity functions in matric variables of any orders >> 1 whose several arguments contain the divisors dy, ds, --- of a set of integers Ny, Na, - ++ and their conjugates, 0, Os, --., namely we have d;8, = ny, d30; = my, -... The doubly periodic functions of the first kind (elliptic functions) do not in general give rise to such parity functions without the use of elaborate reductions by means of the arithmetical theory of quadratric and bilinear forms, all of which are avoided by the simple analysis of the expansions. Accurate forms of these are given in the Messenger of Mathematics, vol. 49 (1919), p. 83. A typical one is ya (zy) = ese y+ 14> i [(— 1 sin (2 tx + vy] ’ in which the first DX refers to » = 1,2, 3, ..., the second to all pairs (¢, ) of integers ¢, = 0, of which = is odd, such that ¢¢ — wn. The same list is also given, with a less convenient notation for the divisors, in the Transactions, vol. 22 (1921) p. 207. The entire literature of the expansions of theta quotients (in the usual analytical forms) seems to have been singularly unfortunate in the matter of misprints; Jacob's collected works however are an exception. From the above 16 can be derived by elementary methods a great number of reduced arithmetical expansions for powers and products of elliptic functions and for the doubly periodic functions of kinds higher than the second. A selection is given in the Quarterly Journal, vol. 54 (1924), pp. 166-176; the few misprints can be easily corrected by means of the outline of the method used. The principal sources for expansions of functions of the third kind, all of the greatest interest for arithmetic when 90 ALGEBRAIC ARITHMETIC. reduced to their proper forms, are Hermite's (Fuvres, the These of Biehler Sur les Developpements en séries des fonctions doublement periodiques de troisiéme espéce, Paris 1879, the papers of Appell in the Annales . . . de I Ecole normale superieure, vol. 1 (1884), pp. 135-164; vol. 2, pp. 9-36, 67-74; vol. 3, pp. 9-42; vol. 5, pp. 211-218, also Acta Mathematica, vol. 42 (1920), pp, 341-347, and the series of papers by Krause, Mathematische Annalen, vol. 30 (1887), pp. 425-436, 516-534; vol. 32, pp. 331-341; vol. 33, pp. 108-118; vol. 35, pp. HT7-H87, also his treatise, Doppeltperiodische Funktionen, Leipzig 1895. Almost any one of these references offers an inexhaustible source of arithmetical theorems when subjected to the methods of this chapter. The introduction of indefinite binary quadratic forms into theta expansions was first effected by Petr, although it is probable that certain theorems stated by Stieltjes were ob- tained by such means. The papers of Petr are in Czech; independently most of the fundamental expansions were found later by G. Humbert, Journal des Mathématiques, 6 ser., vol. 3 (1907), pp. 337-449. Humbert’'s 6 basic expansions are reduced to their arithmetical forms in the Quarterly Jowrnal, vol. 49 (1923), p. 328. These are particularly suggestive in the theory of generalized class number relations. These few indications must suffice, as I hope on another occasion to consider the arithmetic of theta quotients from the present point of view in detail. Such expansions as are necessary for illustrations presently will be stated without further reference. 13. Elimination of negative powers of sines and cosines. Many of the arithmetical expansions contain terms in sec, escz, tanz, cotx, sec(x + y), tan(zx+y +2), --. ete. Such terms must always be eliminated from any identity I between circular functions obtained by equating coefficients of like powers of ¢ in an identity between arithmetical ex- pansions of theta quotients before proceeding to paraphrase. Let ¢ (x) denote any one of sec, csc, tanx, cote, and ¥ (y) either of siny, cosy. Then any product of the form ¢ (x) ¥ (nx), where »n is an integer, can be reduced to the form A¢'(x) + MULTIPLY PERIODIC FUNCTIONS. 91 a finite sum of sines or cosines of integral multiples of x», where ¢' (x) is a definite ¢ (x) and A is independent of x, (4 may = 0), by means of 16 formulas ¢ (x) ¥ (nx), n even or odd, of which typical ones are n Yo cos2nz csexr — cscx—2 sin 2r—1)2, r=1 cos(2n-+ 1)x secx = (— 1)» | + 2 > (—1y cos 270]. p= If a power ¢"(z), »>1 occurs, » applications of the appro- priate ones of the 16 formulas effect the reduction. If a function of more than one variable, say ¢ (z+ y) Yn, 2+ ny), where n,, ns are integers, is to be reduced, we replace the argument of ¢ by a single variable, z = z+} y, getting then 9 (2) Wn —ms) 2 -mny2). On expansion of the ¥ factor by the addition formulas for sin, cos, the reduction is referred to the previous cases. The complete set of reduction formulas for 2 variables is given in the paper Quelques applications - - - a trois termes in the Giornale di Matematiche, vol. 59 (1921). 14. Applications of division of parities. From the examples given shortly many interesting consequences in the same number of variables or fewer may be derived by the methods of Chapter I, §§ 18-21, among which we note the following. Consider for example a relation of the type (3.32) with # = I(x, y, (2, 7)), where the x, y, z, » are independent variables. Since / is a parity function in (3.32), each of the parity functions / (|, y, 2), a ( , y, 7) may replace 7 in (3.32), since each satisfies the parity conditions upon /. More generally, if any or all of the variables in any matric variable to the left of the bar be deleted in the symbol of a parity function, or if in any matric variable occurring to the right of the bar any number of variables less than the order of the matric variable concerned be suppressed, the resulting functions are instances of the original, and such transformations may be applied successively to deduce from a given (3.12)—(3.32) instances of the latter in fewer variables. Hence, the notation being as just after (3.02), we get in this 99 ALGEBRAIC ARITHMETIC. way from (3.12)-(3.32) respectively the following numbers of paraphrases concerning functions whose orders do not exceed the orders o, wy, w; of f, g, h, ow—0, on 0, —0, gu, gm gd, including the case where the functions are constant (parity (010). : Again, further instances can be obtained by replacing the variables in any matric variable by a linear homogeneous function of themselves with integral coefficients, with the restriction that the coefficients be not all zero ‘if the matric variable is to the right of the bar. This includes the in- stances just stated. As another useful transformation, it is clear from the definition in § 2 that any common factor (in the sense of rational arithmetic) of all the integral values of a given matric variable in (3.12)-(3.32) may be suppressed. Sums of functions transformed in any of the above ways, or linear homogeneous functions of them with integral co- efficients are further instances. We note finally the following among the numerous trans- formations leaving the truth of (3.12)-(3.32) unchanged. Certain . of the variables, scalar or matric or both, in any (3.12)-(3.32) may be subjected to all the substitutions of a finite group G' on them whence, by addition, are obtained arbitrary functions which have parity and which are invariants of G. This can be generalized to substitutions which change the signs of the functions, or which reproduce the functions multiplied by constants. 15. Consequences of Jacobi’s theta formula. For the derivation of paraphrases from the theta quotients, Jacobi’s memoir, Theorie der elliptischen Funktionen, Werke, vol. 1, pp. 499-538, is the most prolific source. Its arithmetical consequences have barely been noticed, and not at all in any systematic manner. We need give only a few instances. The first arithmetical identities of the types (3.12)-(3.32) were stated by Liouville, who asserted that he had found MULTIPLY PERIODIC FUNCTIONS. 93 them by elementary means. Proofs for most of his formulas by such methods were given by later writers, references to whose papers will be found in Dickson's History, vol. 2, chapter XI. The entire set of Liouville’s general formulas follow by ‘the principle in § 3 from elementary transformations of Jacobi’s theta identities, and these in turn are implied by Jacobi’s formula for the multiplication of four theta functions. The same is necessarily true of Weierstrass’ equation of three terms, since, as is well known, this equation is equivalent to Jacobi’s formula. The formula itself (see Jacobi’s proof) rests on the elementary identity which transforms a sum of 4 squares into itself, and all proofs by elementary means of arithmetical theorems of the types (3.12)-(3.32) amount to repeated applications of this transformation. It will be of interest to indicate briefly the analytical origins of those of Liouville’s theorems for which algebraic proofs have not hitherto been published. The complete deduction of the theorems from the theta identities and expansions stated is a straightforward exercise in trigonometry. As a first example let ¢ (w, z, u, v) be a restricted parity function (see § 2) subject to the restrictions pw, z, u,v) = —¢g(—w,—z,—u, —v) = —o(w,—z, v, un), the first of which states merely that ¢ has the relative parity p(w, z, u, v)). Then, the > referring to all sets of integers Vi, V2, M3, V4, ¥5 Such that », vy, »,, 7; = 0 are odd or even, #;=0 is odd, and for the constant « = 1 mod 4, a = 421402 p+ 4024402, we have the following, = 1 ls) X ly (2+ 2st ts, vis 2vy, My, VPs) + 9 (2v, +4 29+ ps, — 20, ps—2,, —v1+v,, ¥,1+v5)] = 0. In this (as usual), (—1|m)= (Q1)*D2 In particular we may take f((w, z, u, v)), unrestricted of the indicated absolute parity p(|(w, z, u, v)), and choose 9 (w, ZU, 0) = Sf (w, Zy, u, v)) 17d (w, 5 u)). 04 ALGEBRAIC ARITHMETIC. This is the most general ¢ satisfying the stated conditions. For the theorem is the immediate paraphase of the following theta identity which is easily deduced from those given by Jacobi, [Ole uw, DA Oly, —u, —2)] 92) 50) = [Ds, v, =2) = Olu, —0v, — 2] Sln-t2) 9,0), where Dopo) = So(htu+0) $(—up) 9 (—o). To introduce functions of the second kind into the identity divide throughout by (z+) $y (x—y) SH (w+2) * (w—2) after replacing uw, v by z-+y, x—y respectively. Then gor ( z+y, wt) P@—y—2) IY (w—ax—y) +g (—z—y, w—) FH @x—y—) SH wtzty =gm{ ‘wy, mylyn gm {=%1y, wt hetytahwte—y) = 0 is an identity in x, y, 2, w. The substitution for u, v is merely to enhance the symmetry of the final result. The expansion for Poo1 is poor (x, y) = csc y+ 4 > q" [> sin (2 tet+ey), the outer > refewing to n=1,2, 3, -.., the inner to all pairs (#7, 7) of divisors of » such that n == ts, {, v0, = odd. Using this and Ix (7) == > 7" C0s2vy = 1-1-2 > cos 2 nx (r=0,+1,42,..; n=1, 23,003), we get on comparing coefficients of like powers of ¢ in the theta identity its paraphrase. Write Zio, Y, 2 t) == Jz, Ys {, 0), where fis unrestricted of the indicated parity p(x, y, (2, t)). Then, m being an arbitrary constant integer > 0, and the > MULTIPLY PERIODIC FUNCTIONS. 95 on the left of the CE extending to all sets of integral solutions (im, a 0) of 9 o = > ; 5 m = mi +m i i a wm, m, 20, 4,0dd, d,,4 0, and that on the right to s=20,1,..., a—1 taken over all integral solutions (a, 0) of m = a? b?, a0, the paraphrase is Sr — 2m, ds me— my, dy-+ met my, d+ 2m) = NP (3g —2s—1, a—b, a+b 2012-11). This is equivalent to Liouville’s 16th Article, Journal des Mathématiques, ser. 2, vol. 9, 1864, pp. 389-400. Since both this paraphrase and that first given are equivalent to the original theta identity, it must be possible to transform either into the other by elementary means, but I shall not attempt to do so. Both express abstractly identical arithmetical relations. The substance of Liouville’'s 13th, 14th and 15th articles (loc. cit. pp. 249-256, 281-8, 321-336), is given in the following paraphrase concerning the same # for the partition (mm constant) m = m+ 4m + 2d d, 0 d, 6, odd, so that m is odd, F600, ds + 2 me— my, dy+ 2me+ my, 2%, sm) = ZS FFE, e—ro—1, at 201, HE), 3) dys J, > 0, yy the second 3 referring to 5 = 0, 1, 2, ..., (e—3)/2 over all solutions («, 8) of 2m — «®+ 4° (so that «, 8 are ne- cessarily both odd) in which « >1 and #8 has the sign that makes («+ 8)/2 odd. (See for fuller details Liouville, loc. cit., pp. 321-336). This is one paraphrase of the easily proved fact (which follows without difficulty from Jacobi’s identities) that [y {w, Zz, uw) Y (uw, 7°) Jy (v), 96 ALGEBRAIC ARITHMETIC. in which - |r os >| Y (uw, Z, u) == J Ug (w—u, 7%) I 252); is unchanged in value when u, v, z are replaced by v, u, —z respectively. For, from this we infer at once w 9. (w—u, q%) Poot ul Jy We w . 7 $e (+2, 4°) 93 (w—v, 2) = Yo01 [3 ir , —u) » Se (v — 2, ¢°) Ss (w+ u, ¢?) | of 0 = Poor [* : TEE /) op) (nt a 7°) I3 (w+ v, 7°) a 0, an identity in w, z, u, v which paraphrases immediately into the stated theorem on substituting the expansions and pro- ceeding as before. By means of the transformation of the second order either of these ¢ identities can be inferred from the other, but it is simpler to deduce them separately from Jacobi’s lists of theta formulas. Another of Liouville’s theorems of a similar kind for which no proof has been published is the following, which he used in obtaining several of his class number relations. The odd constant integer mm 0 is partitioned as follows, m= 2m2-4 4, 2m = mids dy, in which d”, 0”, dy, 0,>>0 are odd, m, is odd, and these are all the restrictions on the integer variables. Then it is stated that (loc, cit., vol. 7, p. 42), CT 2w', 0" —2w, 2m +d" —48") es Srl g 9, Ph yal 2% J 2 where F(z, y, 2) = f((y, 2) | x), this f being an unrestricted parity function of the parity indicated. The theorem is the MULTIPLY PERIODIC FUNCTIONS. 97 paraphrase of an immediate consequence of the following special case of one of Jacobi's identities, Ys 91 (b+ 0) 93 (c+ a) So (a4 1) = Y3(a +b +c) F(a) 92 (b) % (c) he Jo (a b ue ¢) BV (a) I (b) Is (0). Change the signs of 0, ¢. Then, by elimination, Po Ss (c—a) [41 (D+ ¢) 0 (b—0a) 5 (b—c— a) + 9 (b—0c) 9 (b—a) Is (b+ c— a)] = S3(a)$(b) 9s (c) [$3 (b + er) Ss (O>—0-+a) + Sob c—a) 33 (b—c +a). Factoring the last by the addition theorems for the thetas we find finally G1 [HD+ 0) 9 (D— a) 5 (b—c-F a) + SH (b—c) So (b— a) 95 (b+ c—a)] == 2%, (0) PH (b) v2 (b) Ja (¢) Is (a) Jo (c— a). By means of the last identity we readily see that 4p t2,9—2, 00) 8: 0—y + 2, 0d + ¢r00(x—2,—y+2,¢°) K@x+y—=z, ¢9)] r+ r—z r—z x-t+z = 5,0 yu00 [© I ew (25, 2 + rue S5E, a + 133 (= 224]. Using the expansions of 09, 9133 as given in the papers cited in § 12 we have the theorem. This example illustrates a useful algorithm. Starting from any of Jacobi’s (or Briot and Bouquet’s) theta identities, we change the signs of one or more variables and eliminate terms unchanged or changed only in sign by these trans- formations. We then use the addition theorems for the thetas to separate sums into products, or inversely. TLast, by the rr i 98 ALGEBRAIC ARITHMETIC. transformation of the second order we can frequently reduce the number of theta factors in several of the products before division of the entire identity by the same theta product to introduce functions of the second kind. There is no difficulty in writing down 4, ¢ identities concerning products of many factors. The desideratum, to reach simple paraphrases, is that the number of factors «%, ¢ in each term of an identity deduced from a given identity shall be a minimum. This is the origin of the several reductions in the above examples. Any stage of the reduction yields a paraphrase; all such are arithmetically equivalent, although this may be troublesome to prove by elementary methods in a given instance. Continuing with the examples we find from Jacobi as just indicated that [Fo(x41) Hh (x+u) So@x—u-tv) H@E—utv) $35x—20) —Jo(x—u) SH (x—u) Sox+u—v) $@+u—v) $3(x+ 20) x Io (10) Gy (20) is invariant when wu, v are interchanged. Putting u = yz, = y—=z we get after a few simple reductions P111 ~ ily py 9 eet Y > 9 Ln Punks od fbr ’ 3 = = == 0 Zr 0 YZ Na jo—2y—22 5 : = ) olk2 —y—r og 2322s oT LE =0 —puus [2 ’ ge an identity in x, y, z, which paraphrases at once into Sr -tw, o'—2u, 2d" + 2m'— 0") = &(m) fe Fm? 5, 5) —2F t, 2m2, 28), where F' is the parity function Fle, yy, 0) = flim, yu, 2%; MULTIPLY PERIODIC FUNCTIONS. 99 the > on the left refers to the partition 3 m=", wEQ 4, °>0, for m fixed (the integers m, m’ d”, ¢” are without further restrictions); é(m) = 1, 0 according as m is or is not a square; 1 =; Yeler respectively to §= 1,2... 2912-1, t=1,9 00 plod, This is the substance of Liouville’s 12th Article, loc. cit. vol. 5 (1360), pp. 1-8. Completely arbitrary single valued functions of integers (not restricted as to parity) of one or more variables can easily be obtained by the same means as above. But it is to be noticed that, so far as general formulas are concerned (those containing the arbitrary functions), this is no essential gain in generality over the parity paraphrases. When we come presently to the theta functions of » > 1 variables we shall see that the arithmetical equivalents of theta identities always involve completely arbitrary functions. One example for the elliptic thetas will suffice. The pseudo-generality is attained by the device of adding an identically vanishing sum to a parity identity. The expansion of 91/90 (1) in the reduced form is given in the papers cited in § 12, : —0 $90 (ur) = 2 git [Zs 10) cos (4 5 J] the outer XY veferring tom = 1,5, 9, 13, -.., the inner to all d, d >0 such that m = dJ. Hence the identity So (1/2) - 91/90 M/2) = I; paraphrases into CENCE where m = 1 mod4 is constant, and m = 4»? d,0 » = 0, dy, 6,>0. Now, for suitably chosen f(z|), 7(|«) — nr r= = &(m) (—1| mW" m'2 £0), | i 100 ALGEBRAIC ARITHMETIC. the arbitrary on) = fle) f(z). Hence if it can be shown that over the same partition £3 le 0) f 1 4 nA == 4, we shall have proved that Zr 18 nk BGR) = etm uy 0). By the principle of paraphrase it evidently suffices to prove (A) when f(|z) = sinzf. Expanding sin (v al a t by the addition theorem we see that it is sufficient to prove, for a particular »,, that (# is a parameter) N (—1|6)sn (25%) == 1, But this is obvious, since to a given term in the sum corre- sponds a term with d,, J, interchanged and, for m = 1 mod 4, d: 0; = 1mod4, whence (—1|dy) = (—1]|d;). Thus the ¢ identity is proved. This was stated by Liouville, 11th Article, loc. cit. vol. 4 (1859), pp. 281-304. The other main formula (mr) of this Article is the paraphrase of an equally obvious theta identity. Expressing So @+2 (x —2y)+ %H@—2y) 9 (x+2y) as a product, multiplying both sides of the resulting identity by 91% (y, 4) Is (y, q¥?) and dividing out by 95 (x) $0 (2), we get [29250 (z, 20) 93 Qy — 2) + goo (x, — 23) 93 2y + 2)] = 9230 (y, Y, 7% I (x) -|- $230 — Y, — ¥, 79 Jo (x), which gives Liouvilles’s (7) as stated (the explicit form will be found in Dickson’s History, loc. cit.). These examples will suffice to show that even trivial theta identities yield interesting results. We have chosen known theorems as illustrations to suggest the ready means of proving MULTIPLY PERIODIC FUNCTIONS. 101 or disproving any such formula which may have been stated. Moreover from the theta identity giving the algebraic proof an indefinite number of further theta identities can A: COD | structed by multiplications, resolutions into factors. climinations, ’s ete., from which additional paraphrases can be obtatned:: We >> 30 00! give no examples of the specific arithmetical theorems which such parity identities imply upon specializing the parity functions, as we are concerned primarily with general methods. If no shorter way of tracing the theta origin of a parity theorem suggests itself the following will always lead to the equivalent identity between theta quotients. Replace the given theorem (3.12)-(3.22) by its special case (3.11)-(3.31). Separate all trigonometric products into sums. Rearrange the arguments in these sums as linear functions with respect to the x, y variables. By inspection it is then easy to write down the product of thetas and ¢’s (or theta quotients) which generate these terms, on glancing also at the partitions in- volved. The resulting theta identity is then to be proved from first principles by means of the classical relations between the functions. The reverse process is of course the natural one—start from theta identities and deduce paraphrases. In all such work full lists of theta expansions are necessary. 16. Application to class number relations. We take space to indicate only two of the methods by which para- phrases lead to class number relations. The first was pointed out by Liouville, see H. J. S. Smith, Report on the Theory of Numbers, Art. 136, or Dickson's History, vol. 3, chapter VI. It consists in choosing for f(z) in a given paraphrase arising from a cosine identity its instance ¢ (x), where ¢ (2) = 1 or 0 according as |x| = ¢ or |x| >¢. This leads to identities concerning numbers of representations in quadratic (binary) forms; at least one enumerative function thus encountered will in general involve one or other of the class number func- tions #7, F or a linear combination of them, F-+ F, — @, ete. The second method applies those arithmetical expansions for theta quotients which directly involve F, F}, G, -... The derivation of such expansions ultimately depends upon the 102 ALGEBRAIC ARITHMETIC. classical theory of binary quadratic forms. Examples of the last will be found in Humbert’s memoir cited in § 12. Assuming . that we possess several such expansions we can apply pro- ecspes. similar to those of § 15 to deduce an indefinite number “of paraphrases. These will involve the class number functions and parity functions. By specializing the latter we obtain any desired number of class number relations. Both methods can be used with profit simultaneously. An application of the first method to a given paraphrase suggests reductions in obtaining expansions useful for the second. A fruitful special case of f(xz|) in these connections is f(z) = 1 or 0 according as z = 0 mod is true of false. An extremely prolific identity for class number relations is the following, due to Professor J. Ouspensky (references will be given presently), who proves it by elementary means. For the partition m = »24+do of the arbitrary constant integer m 0, where », d, J are integers, and » = 0, d, 0 >>0, he shows that Nieto, v,d—8)—2¢00—2v, d+, 2d—8- 20) = e(m) 2X [yp @m12, mY2—j, 2mi2— 25) — mj, mi, 2m), where ¢ (z, y, z) is the parity function ole, y,.9) = f(y, 2x), z(m)y == 1 or 0 according ag = is or is not a square, as before, and the sum on the right refers to; =1, 2, -.., 2m!>—1. This formula can also be proved from the identity 1 | as bs Cs | as by cy [= 0, | sy bs Ca | on taking % = 32x +y+29), & = 9@0s—y—29, g = ¥4;(y) (= 2,0) MULTIPLY PERIODIC FUNCTIONS. 103 whence om et 2, —z, 45 93%) = pu (@—2, y+ 22 4% @e—y—20) + yu (@ 2, y—=2z 1?) 4 ety 27), from which the theorem follows at once by paraphrasing. The fundamental character of the ¢ formula is amply demonstrated by Professor Ouspensky’s applications. Leaving class numbers for a moment, we may call attention to one of these in another direction. An instance of ¢ (x, y, 2) is clearly (—1|2z) (—1)Y ¥(y, x) where ¥(y, x) is the parity function f(y, x), provided z be an odd integer. It is readily seen that the ¢ identity gives for this choice the following, 2 (—1|Ow(d+v,0—20) = em) (— 1)" "1 3 (—1 |j) @ (m2), where 0 is now restricted to be odd. By choosing for vw its instance v(x, y) — xy, the classical theorem concerning the number of representations of m as a sum of 2 squares is obtained; similarly from ¢¥(z, y) = 2% and ¥ (xz, y) = zy° the like for 6 squares follows; for 10 squares there are three sich choices, 2%, 2%° ‘=f, and so on, Tor 4, 8, 13, ... squares we start from other immediate consequences of the fundamental formula, and by equally simple specializations obtain the numbers of representations. The application in all cases proceeds from the following easily proved lemma, (Bulletin del’ Académie des Sciences del’ URSS, 1925, pp. 647-662, where the proofs of the theorems indicated above are given in detail). Let Np(m) = the number of representations of m by a sum of p squares, and let @(m) denote an arithmetical Junction defined for m =0,1,2, .... Then, if OQ) =1, and 2m—(p+ fl om—r) =0, »=0 £1,132... the sum continuing so long as m—v® = 0, this implies Ny(m) = @(m). We thus have a new approach to the problem of enumeration of representations as sums of squares. 104 ALGEBRAIC ARITHMETIC. Returning to class numbers, we can only indicate the power of Professor Ouspensky’s identity by stating a few of his applications. First, by easy specializations of ¢ and com- bination of the results he deduces a number of related general formulas, all of importance in class number relations. It is an interesting exercise to follow the evolution of these identities (and other of a similar kind) from elementary al- gebraic transformations of the equivalent theta identity which generates the general formula. The author then determines (by the method first used by Hermite in proving Kronecker’s class number relations, indicated at the beginning of this section) the numbers of solutions of certain indeterminate equations of the second degree when the indeterminates are linearly restricted, for example 4n-+1 = dé-2d4d'¢, where 4,0, d’, 8’ are integers >0, 24 = d--4-£2, and ¢ is odd. In the algebraic method such results are most easily reached by taking f(z|) = 1 or 0 according as |x| = ¢ or \z| 4 ¢, where ¢ 0 is a constant integer, in the simplest paraphrases deduced from Humbert’s expansions. Among the class number formulas which the author proves by these strictly elementary methods are the classical eight of Kronecker, those of grade 2 of Stieltjes, and those of grade 5 of Petr. A class number relation is said to be of grade % if in it the arguments of the class number functions decrease by integers of the form k/4* where the several 7 are in arithmetical progression. The first derivations of such relations were by applications of the whole machinery of the elliptic modular functions; here they are referred to elementary arithmetic. Similar derivations are given of relations of the Hurwitz and Humbert types, and those of grade 3 of Petr, also of Liouville’s. Finally the relations of grade 5 due to Gierster and Chapelon are obtained, and the method is capable of indefinite extension. Incidentally many new relations are indicated in the course of the derivations. These memoirs are a striking example MULTIPLY PERIODIC FUNCTIONS. 105 of the great richness of paraphrases obtainable from the elements of elliptic theta quotients. They are published in the Bulletin de I Académie des Sciences de UURSS, 1925-6, in SIX memoirs. To illustrate the second method of deducing class number relations from paraphrases we take the identity Jo 2S: (2) So) Siz) = = [9 S31 (2) So (x)] - [2 Y1(2) Fo (2) So ()], the function on the left being that whose expansion introduces the class number functions. For if as usual F(») denotes the number of odd classes of binary quadratic forms of negative determinant —»n, we have for the expansion of the theta quotient on the left ° 23 go 2Z Plo — 1 cos bir — > (0 — qd) cos mi | where the first X extends to « = 1,5, 9, 13, ; --, the second to all odd integers $==0 such that «— 4%*>0, and 3X’ to all pairs (0, d) of divisors > 0 of « such that d<0; that of the second is ZT sin (© 5 i (5 =108.711 15+.) where > refers to those divisors d, d>0 of 8 such that B= dd, dd. The paraphrase is therefore 3X Fla=0 ) f(b) eri the > on the left referring to & as above, and those on the right to all positive integers defined by 106 ALGEBRAIC ARITHMETIC. 0 25, vo =4d48, 4d<8, m =v, B= d:0 where the constant integer « = 1 mod 4, and 3; = 3 mod 4, all d, 0, d;, ds, 7, (the last necessarily odd) satisfying these conditions being taken; f(x) = the parity function f(x). By specializing f(x) an infinity of class number relations are obtainable, and all these are equivalent to the given theta identity. The applications of the parity considerations re- garding forms in §§ 6—11 are particularly interesting, but we need not stop to write out examples, several of which are given elsewhere. Paraphrases involving functions of order w may be called w-fold infinite, for an obvious reason. The above example is a singly infinite class number relation. It is not difficult to obtain w-fold infinite "class number relations for = 1, 2, 3, 4 from Humbert’s expansions and Jacobi’s formulas. From all the examples given the general significance of the orders and degrees w, y, ,, d, dy, 9; of parity functions in § 3 and its relation to the theta quotients giving rise to the respective paraphrases is evident. APPLICATION TO THETA FUNCTIONS OF p>1 ARGUMENTS, §§ 17-20 17. Arithmetical expansions of the theta functions of p>1 arguments. In one of the customary notations for these functions we have the expansion (for conditions of convergency see Krazer, Lehrbuch der Thetafunktionen, or Harkness and Morley, Theory of Functions), 5 1 = {2 gs - . Ip Tr cee vp) eh hy hs -- = — 00, © LH & 1; = Y ap =| 3 3 rs (r+ 5 90) (ust 0) Myron Wy, » ee 2 > (n, =) (vr + 4 ml, r=1 = with 7, = 75, and each of g;, A; a definite one of 0,1. MULTIPLY PERIODIC FUNCTIONS. 107 Define Viy Upy Gry = ++, by 2% +g =v, avr =u, XplUnr/A)=q, r=1, --:, 0), WE [h?] = exp [Zr > hv), 2 ~~ Rr rbs = 2p pbs (r =°1, 2, ....p—1; 81,2, ..., p—7) Then [hv] = 0, 4, —1, —4 according as > hv; = 0, 1, 2, 3 mod 4. The above expansion becomes ? =1p—7» 3 (/iv] qn qs oe a exp > Voll + =. ~ Vy Vsiy, hod) | the Y referring to nn, = —w to.» (r = 1, .-:, yp). The »’s in this are integers, the u’s independent variables, the q's parameters. This is the form of the expansions required in arithmetic. We shall require the following lemma, which is proved immediately on separating first reals and imaginaries and observing that in a trigonometric identity involving tri- gonometric products of different parities all those terms having a given parity may be considered independently of the rest in paraphrasing, and finally recombining terms in an obvious way. If 8, z are wmotric variables of order wn, ond & (= 1, ..--, ©) are t integral values of &, then if t 17.1) 2 ajexp [ig] = 0 J=1 is am identity in z, where the a; are rational numbers or complex numbers of the form bj—-ic;, where by, ¢; are rational numbers, the identity implies rein =0 Yan 20 =0 and hence it implies where f(z) takes a single definite value for all integral values of the matric variable z of order mn, and otherwise is entirely arbitrary. Thus identities of the form (17.1) lead to arith- 108 ALGEBRAIC ARITHMETIC. metical theorems concerning functions wholly arbitrary except as to uniformity for all sets of integral values of their » in- dependent variables. 18. Quasi even and odd arithmetical functions. When several [hv] are considered simultaneously we write p [ery] = [Arr vrs hop var, «--, Tope Vr] == > bev, /=1 and the value of [%,v,] is 1,¢, —1, —¢ according as the sum just written is =20, 1, 2, 3 mod 4. In what follows each of Yrs Lejy G=1--,p;r=1,2, 3, ..., 2%) denotes a definite one of 0, 1; each nj. is an integer (positive, zero or negative); », — 2n;.+g¢;~ The [/,»,] are multiplied according to : Verve] [Des vs] == [rr v1r + Mis?is, hoy vo, + Tos vas, riviey Ppr Vor + igs Vps). The characteristic . gril. [91r Gor Gor 1 Ye == ( 2 3) JZ | I hoy Sq = p is called even or odd according as > gjrhj- is even or odd. j=1 Prom the v;,(; = 1,..+, p) and their products vj, vi, (j,k = 1,-..,p,j +k) we construct the function [Ar vy) L (v1, Vary === Vpry V1 Vary »++, V1rVpry VorVar ++oy Vp_1,r Vr) where L is a function of p(p--1)/2 arguments which takes a single definite value for each set of integral values of all its arguments, and which otherwise is arbitrary. The product [/iyv,] L( ) just written will be abbreviated to either of the forms Gy ,) (18.2) |: | I) Oy as convenient. This function has by definition the characteristic (18.1) and /(»,) is called quasi even or quasi odd according as its characteristic is even or odd. The function obtained MULTIPLY PERIODIC FUNCTIONS. 109 from a quasi even /(».) on replacing each of the first p arguments in L by zero is denoted by either of Ir @,) ? (18.3) J [2] Sin and is called a quasi constant. By the usual theory of theta characteristics (which can be transposed verbatim to these functions) we see that there are precisely 27—1 (2? + 1) quasi even /(»,), and hence the same number of quasi constants, and 2771(22 —1)o0dd/(»,). The paraphrase of identities between thetas in p arguments leads to arithmetical theorems concerning functions (18.2), (18.3) for sets of integral values of the arguments obtained from the simultaneous solutions of a set of indeterminate equations. 19. Products 7, 2. The paraphrases refer to certain products, in a technical sense, next defined. Let # > 1, v = 0 be constant integers, and »,.(r = 1, ..., p) arbitrary constant integers >0. For the »,; as already defined con- sider the set of all (integral) solutions »,; of the p equations +7 (19.1) n, = Sa (rm= 1, eer, PD), ji=1 Set t+r t [7] =] wn» a, = r=1,2,< 0) J=1 I= +7 Bris = > Viy¥ris; =1,-v.,p—1:8=1,2,..:.9p—7), Jj=1 and for each solution of the system (19.1) construct the function [7] Ley, Oa, =», Op, B12, B13, at Bip, Bas, St Bap grits By—1,p)- Take the sum of all values of the last over all (necessarily finite in number) solutions of (19.1) and denote it by 1) 10D) I) «+ TEA Brin) A (Begs) - + A (ep). We call this the /, 4 product, or simply the product of Hn), 4ilvy) (== 1, vse. Gils tela. ith), 110 ALGEBRAIC ARITHMETIC. Such multiplication is commutative. If the parameters (ny) = (ny, Ms, - --, my) are understood we omit the (n,)| and write AW) = 1): Ur) (egn) s+ A {rir). If + = 0 the factors 4 are absent. Clearly #(») is uniquely determined by (n,) and the characteristics Yr 12] i me < [oh | Jus ( 1,0 ¢ Id,4:0,7) of the /(»,), 2(vs) respectively. Hence if preferred 7 (v) may be considered as a function of (n,) and these characteristics. In determining the degree of a product -#(v) we treat the quasi constants as absolute constants. The degree of A(v) as above is therefore . Hence a linear homogeneous relation between products of degree ¢ is defined. 20. Abstract identity of the theory of the p fold thetas and that of /, i products. Let (20.1) 24,0) be a homogeneous linear relation between products of degree ¢, and let the characteristics of the /, 4 in 4; be gs (20.2) I G == 1.2, -.+ 5 7=0,1,..., m). Consider the theta functions of p arguments v, having the characteristics (20.2) and in the last = of these put v, — (r = 1, ---, p). Denote the (common algebraic) product of the ¢ theta functions and = theta constants thus obtained by ©;(v). Then (20.1) implies n (20.3) ; 2k 6) = 0, j=0 and conversely, (20.3) implies (20.1). For, from preceding sections, it is easily seen that the result of equating to zero the coefficient of ¢;* ¢;--- ¢ in (20.3) is the special case of (20.1) in which each function L ~ MULTIPLY PERIODIC FUNCTIONS. 111 is replaced by the exponential function of / (= (— 12) times the scalar product of the argument of Z and the matrix of order p(p-+1)/2 whose elements are v's, u's in § 17. By the lemma in § 17 the converse above stated follows; from the converse follows the exponential form of the theorem and thence, by multiplying by ¢}* ¢,* - - - gq and summing with respect to ny, ny, ---, nn, We get (20.3). Hence in any identity between theta products (such as, for example, the classical biquadratic relations) the theta products may be replaced by 1, L products having the same respective characteristics as the thetas, all such products being taken over the same (ny). Thus the algebraic part of the theory of the thetas is equivalent to sets of identities between arbitrary single valued functions of integers satisfying sets of indeterminate equations of the second degree. The pseudo periodicity can easily be traced to the transformations which such identities undergo when the numbers of odd, even variable integers in the sets of equations are changed. By means of the parity forms developed in §§ 6-11 the set of indeterminate equations of the second degree can be replaced by equations of arbitrary degrees, or such extensions can be reached directly by im- posing on the arbitrary functions in (20.1) the condition that they shall vanish except when one or more of their arguments are integers representable in any preassigned forms, and similarly for other arithmetical restrictions, as, for example, that the functions shall vanish for integral arguments that are residues of assigned powers. This does not of course permit us to replace (19.1) by a set of conditions given at random; the manner in which (19.1) still dominates (20.1) even when (19.1) is replaced by a set of equations of degree = 2 will be evident on working through any special case, say that of the thetas of 2 arguments. CHAPTER TV APPLICATIONS OF THE ALGEBRAS €¢, © ALGEBRA €, §§ 1-9 I. Scope of KE. What follows is a short introduction to an extensive branch of algebraic arithmetic, namely to its multiplicative aspect. In this the algebra €, a species of resultant of €, 9D, plays the central part; € is an algebra of unique multiplicative decomposition. The basic concept is that of composition of matrices. Although addition and subtraction are defined in €, and are abstractly identical with the like in 2, they are of less interest than multiplication and division, and their detailed development may be omitted. Multiplication and division in € will be placed in (1,1) correspondence with the multlplicative properties of functions of two independent variables in 2. From relations between such functions we infer without computations relations between functions of any elements for which unique factorization subsists. When the functions in 2 are restricted to be rational and integral, the correspondence gives an arithmetic in € abstractly identical with rational arithmetic. As € or its special arithmetic has great power over the algebra of the great mass of existing arithmetical functions, and as it is also extremely effective in devising new functions and classifying their algebraic properties, we shall state its processes in detail. From this several possible generalizations will be apparent, particularly to the properties of sets of functions of elements for which the fundamental theorem of arithmetic holds. The initial treatment is abstract, as € has been con- structed to include a wide range of existing algebraic relations between arithmetical functions and to make possible indefinite extensions in similar or essentially new directions. 2. Composites. Let u;(G = 0,1,.--) be a set of ma- trices, finite or infinite in number, each of finite or infinite 112 APPLICATIONS OF THE ALGEBRAS €, 9. 113 order, whose elements belong to a commutative semigroup © having a unity. Then if each element of yu, is the product in © of one and only one element from each of ux (k = 1, 2, ...), to is called a composite of py, pg, ---. Such a composite is uniquely determined only when the law of selection of ele- ments from the wx to be multiplied together to produce elements of w, is specified. The algebra € is defined by such a law when (and only when) all of the matrices uw; are of infinite order. The following theory can be generalized at once to any set of elements closed under any operation which generates from any pair of elements of the set a unique element in the set. Composites are then defined with respect to this operation which replaces multiplication in &. Thus, for example, there is a theory of additive composition of matrices whose elements belong to a module or to a finite field. 3. Primary and derived matrices. Let s be any matrix in © of infinite order, 3.1) § = (51, Say 2) whose first element s; is the unity in &, so that s,s, = s, (nm = 1,2,.-.). Let p, be the nth rational prime, 1 being counted the first, 9, = 1, py = 2, p= 3, 9, = 8, --s. In (3.1) make the following change in notation (3.2) 8 = Zp, G=12..7, and write (3.3) Ai = {z, Tay Xgy T5y X74 L115 X13 L17y * * J, so that x = s, and x is an element of x only if %. is prime. We call = the primary form of s. Any matrix in & with first element the unity in © can be written in primary form. For distinct matrices s, r, --- in © we may use different new variables z, 7, - -- (with prime suffixes). Let a, b, ---, c be rational integers > 0, and let z,, Lgy +++ Ty be distinet elements of » in (3.3). Then o,8,-::, 7 are 8 114 ALGEBRAIC ARITHMETIC. distinct primes =1. HH a =D =... = ¢== 0 we define xy x ce x, to be the unity in &, so that the value of this product is s; = x;. Let each of a, b, ---, ¢c be now a rational integer > 0, and a, 8, -.-, y distinct primes >1, and let (3.4) Nn = a? 0... be the resolution of » >>1 into powers of distinct primes > 1. Then we define the element xz, of & by JL — S| (3.5) Z = ala)... 2 y i n %.2 Hence a, (im = 1) is a uniquely defined element of &, and the D matrix (3.6) z’ = (om, Loy Lgy Lay ***y Ty * * )) in © has as elements all products in & of positive integral powers in © of elements of the original sin (3.1); moreover the first element of 2’ is the unity in &, and 2’ is a uniform function in & of s. We shall call =’ in (3.6) the derived matriz of the primary x in (3.3). We have therefore defined in any commutative semigroup & having a unity the primary and derived matrices (3.3), (3.6) of any given matrix (3.1) whose first element is the unity in &. Taking now =’ as the given matrix in € we can form its primary, and from the latter, by forming the derived matrix, we obtain the second derived matrix z” of the primay =, and so on indefinitely. In what follows we shall attend only 6.7, 2. Addition in © is without meaning. We next construct from all the elements of © matrices of functions, on the hypothesis that a certain ring of such functions exists. 4. The modified ring Rg of S. A sét 3’ of elements is said to form a modified ring if (1) 2’ is a ring, and (2) in 3’ there exists an element having with respect to multi- plication the properties of unity. ; Consider now the set 2 of all primary matrices and their (first) derived matrices in ©. Let w be any one of the APPLICATIONS OF THE ALGEBRAS GC, 9. 115 primaries, and let we, be any element of w, so that « > 1 is prime. We shall now assume that the set of all symbols Jae); ga(we), ha(we), - - - (a= 0,1, ...; a prime >1) are the elements of a modified ring, Rs, in which each of owe), go (wa), ho(we), -+- (a prime >1) denotes the unity, and in which addition, multiplication, and division by the unity, are indicated as in A. For example Ja (We) ga (we) denotes the product in Rs of f, (we) and : Ja (We). Division in Rs is defined only (by the above) when the divisor is a product of powers of the unity in Rs. It is not assumed that multiplication in Rg has the same interpretation as in ©; in general the interpretations will be distinct. By the definition of a ring it follows that each of f, (we), ga (we), - - - is a uniform function of w. We shall refer to the fi (we), Ja (We), +++ as uniform functions. From Rs select the set of all uniform functions having as arguments a particular wg, say xz, where d >1 is prime. Arrange these functions in any way into matrices in each of which the first element is the unity in Rs. Without loss of generality we may assume all these now to be C matrices. Let (4.1) Jo ee (fo (zy), Lig), ful), +) be any one of them. Then fj is called the primary matrix in Rs with base (fo, fr, ++, fu, +) and argument zy. From the definition (see the remarks on functions, chapter I § 8), it follows that f; is a uniform function of its argument and base, for when xz; and the base are assigned, fj is uniquely known. Keeping the base of f; fixed and letting J range over all primes «, 8, 7,+--,90,..., >1, we obtain the set (4.2) Tos Bs pws » voy vein 8% 116 ALGEBRAIC ARITHMETIC. of all primary ps in Rs having the given base {for Sis : +s Jos Bn Each element i Js is of the form f(x), where j => 0 is an integer and 0 >1 is prime. To define f;(x,), where n is either prime or composite, we note that (43) fo @) filag) «fo (ay) is in Rs. With n as in (3.4) we write the uniquely de- termined element (4.3) of Rs in either of the forms f.(n), Julz), so that 44) fn) = fale) = ful) fo (25) - - Je (z,) (n>1), and to conform with this we write (4.5) 0) = 10) = Hl), Hence (4.4), (4.5) define f»(n) = fu(x) for every integer n>0, and the set of all 7, (x) (n= 1,2, ---) uniquely determines the I) matrix which we call the derived matrix in Rs with the base (fo, fis ++), (see (4.1), and the matric argument as in (3.6). Algebra € is concerned with the laws of combination of derived matrices in Rg having the same matric argument z’ and any bases. One extension of € discusses the like when the arguments are not necessarily the same. 5. E composition. Consider the elements fi (n) = f,, (x) of the derived matrix (4.6) and suppose n>1, so that fr(n) may be taken as in (4.3) with % as in (3.4). Then each of the primaries fe = (fo (zp), Sil)» 9) (0 = a, Brent) contributes precisely one factor to f.(n), the respective factors being fu (x,), fo(zg), - - +, Je (x) and each primary fo, with ¢ different from each of «,8,..., 7, contributes as APPLICATIONS OF THE ALGEBRAS €, 9. 117 factor to fr(n) its first element f; (zo) only, the last being the unity in Rs. Hence f in (4.6) us a specific composite, which we shall call the E composite, of the set (4.2) of all primary matrices in Rs having the base (fo, fi, «+5 Jny ++), and we shall write or in full, (5.2) LG), le, os Fala), 9) = E(fo(@yp), fi(@g), ++, fu(@g)y +++) where 0 denotes the particular general prime >1. It may be anticipated that abstractly £ composition is a multi- plication as J ranges over all primes 1, and this we shall see is the case. The base, we recall, of the C matrix on the right of B.D Ig (7, fi, «La, ~+-) Let Ui, ky, cee ben, et) % == fy SRI h) be a set of bases, not necessarily distinct. Then, as in (5.1), we write the £ composites (5.21) Jf : E(fy), 9g = Egy), FE ho= E (hy), and, as in (4.1), the primaries (5.3) ky BUR (ko (xp), ky (zy), cee fen (24), is 2), (k =f Gy tity h). Take the C product (see chapter I, §§ 17,20) K; of all the ky in 5.3), (5.4) Ky = Po {/5 Ios: hy} me (Ko (x), K, (4), 3 =), and note that the same argument x; occurs in all the factors Ja» 9s +++, hy of this product. The multiplications and additions by which the elements K,(zz) (n = 0,1, -..) of K; in (5.4) are generated from those of (5.3) are in Rg; the indicated C product Pc in (5.4) is C multiplication with re- 118% ALGEBRAIC ARITHMETIC. spect to Rs. The notation Kj, is consistent with what pre- cedes, since each element of the C product is a uniform function in Rs with the single argument xz, and by the definitions K,(z3) = the unity in Rg. Hence K, is a primary matrix in Rg and therefore we may form its € composite E(K,), say (5.5) K = E(Ky), = E(Pc(fs, 95 +++; 2) Observing now that f, g, ---, 2 by (5.21) are E composites, and noticing by (5.2) that an E composite is a D matrix, we can form the D product of the & composites f, ¢g, ---, I, say K’, 6.6) K = Pplf 9, --- BN} = (Ki), Kila), +) From the definitions now follows at once the extremely useful consequence (5.7) K'= RK or, indicating the operations more fully, we have (5.8) Pp (E( fy), E(g,), The E(hy)) Sa E(Pc(fy Jos Tots hy). 6. The four fundamental operations in &. We can now summarize € in concise form. The elements of € are all the derived matrices of Rs with the same matric argu- ment 2’ (see (4.6)). If f, g are any elements of €, J == (f1, (2), Jz (x), a 2), g = (91 (x), J? (a), ? “), their / sum, written fg, is defined to be identical with their J sum, that is, Steg = (h@+nk), rx) +g), --); their &J product fg is identical with their D product, and the unity, 5, in I is the derived sequence APPLICATIONS OF THE ALGEBRAS €, D. 119 nN = (na (1), 12(2), we ) = (ne (x), N2 (x), La 2) in R, where 7,(n) is defined by 72z(1) = the unity in Rg, 52(7) = the zero in Rs, 2 Hence it follows that /# subtraction and division are the corre- sponding DD operations. Algebra © is therefore the instance of algebra D in which the elements are the set of all derived matrices of Rs with the same argument x’ together with the D sums and differences of these matrices. The specific character of €, which distinguishes it from other instances of IO, resides in € composition, whereby any element of € is decomposed (as in (5.2)) into an infinite number of primary factors; the fundamental theorem is con- tained in (5.8). 7. The algorithm of ¢. Since € is an instance of ® we may replace operations upon elements (matrices) of € by abstractly identical operations upon their DD associated func- tions. Incidentally this at once suggests an n-fold generali- zation of €. By a mere change in notation, replacing the letters s, z by r, t wherever they occur in § 3, we obtain from an initial natrix r = 0, 7s, - Yn Sits derived t == (4, 25; ++, Ln, 204); corresponding to (3.6). Hence t,t, = t,t, — f3, 0 any prime. Precisely as in ® we treat the elements of ¢ as parameters. Take as the D associated function f;(¢) of the primary fj in (4.1) (7.1) Fo) = 27 6 falas). Then, since ¢, = t, 3 -+« #5, for m as in (3.4), we see from (5.1) thai (7.2) 1.2 gn) = 3S 0.0.09, 120 ALGEBRAIC ARITHMETIC. the [ J; indicating the D product as J runs through all primes > 1. Either the product on the left of (7.2) or the sum on the right may be taken as the associated function in € of (5.1), for if the left be distributed and rearranged with respect to the parameters t,(n = 1,2, ...), as prescribed by D, we get the right identically. The function associated with the C' product Kz in (5.4) is (rg) 20 Sn) | Gong) LE Ll (ns) = 27h Kana), in which, by the definition of C multiplication, (7.4) Kn (zg) = ~—t (zg) Gv (zg) A he (4), the > referring to all sets of integers a, b, .--, ¢ each >0 whose sum is #n. With the D product KX’ in (5.6) is associated LY nn ( X7 0.G (7.5) == — i Kn), where, by the definition of DD. multiplication, (1.6) Ki) = 2 fr @) foi®@) --- fi), the > referring to all sets of integers p,q,.--,7 each > 1 whose product is n. It follows now from (7.2) that 70 NH. 3 axel =" E06, the sign of equality having the significance explained in connection with 0 associated functions: after distribution of the left and rearrangement with respect to the parameters, the coefficient of #, on the left is equal to that of #, on APPLICATIONS OF THE ALGEBRAS €, D. 121 the right. Thus (7.7) is merely the formal equivalent, in terms of associated functions, of the matric equality (5.8). 8. Generators. We shall now replace multiplication and division in € by abstractly identical operations upon functions of two variables ¢, & in 2. Let ¢, £ be parameters in A, and recall that the zero, unity in 2 are written 0, 1, also that sums, products, etc., are written without special notations, thus t-§&, {§, ete. In the typical factor Dts 1g) of the product in (7.2) replace ¢; by {, x; by & and all operations in Rs by the corresponding operations in A, so that (8.1) Pl, 8 = vy) is a function in A. By convention we assign to f,(&) the value 1 (= the unity in A), and call F'(¢, &) the generator of the derived matrix in Rs. The following (8.2) Fa, 5-07, oo I.E will be read “F(z, ¥) generates f’. If F.I'.f and G.I.g we define the generators F, G to be equal, FF = G, only when f=y¢. It followsthat F=G implies the equality (in A), (fo (5), Ji (8), Aa Jn (&), i 9) To (90 (8), hn (5), an gn (8), pet ). let f,9,--+, 4, --- be the set of all derived matrices in Rs having the same argument; their respective generators are F(;, 5, GE Y, :--, HQ, 8), ---. Then, with respect to C multiplication the set of all generators F,G,..., H is an abelian group ®. Denoting multiplication in & by juxtaposition, thus FG... H, we see that FG -.. H generates the FE product 192 ALGEBRAIC ARITHMETIC. J+ hh; and if F be the inverse of TF in. ®, then F-1.I. f=, where f= = y/f is the E reciprocal of f. Hence the properties of elements of € with respect to mul- tiplication are implied by the like for &, and may be written down therefrom by replacing the symbol of each generator by the derived matrix in Rs which it generates. In applying the last theorem it is necessary to have a short way of getting the derived matrix f generated by a given F(t, §) and also of constructing the generator F(t, §) of a given f. For the first it is sufficient to determine f(x) when F'(¢, §) is known. Write (8.3) Jn (x) Se Ja (z,) Jb (23) 0 Je (x), where (as before) n — a? 8°... y¢ is the resolution of » into prime factors. Since F' (¢, & is given, so also is the explicit form of 7, (8). In fu (9 put (n, § = (a, 2a), ---, (¢,z,) and get fu (xa), ---, fc (z,), and hence f(x). Conversely, to find F(, § from (8. 3), replace (4, xe) in fy (xs) by (n, ¥), multiply the resulting 7,(5) by # and sum for n==0 io wo; then F(t, & = 2° t"fu (8). Note that the first term of any generator ¢s the unity, 1, in A, 9. Arithmetic in € The derived matrix f is said to be algebraic or transcendental in € according as its generator F(Z, &) is an algebraic or a transcendental function of ¢,& in A, and the corresponding generators are similarly named. To include numerous special theories we now frame a definition of fundamental domains. Let the £;(§) (j= 1,..., nm) be algebraic functions in A, where # is a finite integer > 0. Then FUD=1150+ 0H 0 is a polynomial in ¢ and F (f, &) is an algebraic generator. Let ¢ +0 be in A. Then the algebraic function F(z, §) +c in A is not a generator, since the term independent of ¢ is +1 (the unity in A). An algebraic generator, such as the APPLICATIONS OF THE ALGEBRAS CG, D. 123 above F(t, §), which is a polynomial in ¢ is called fundamental. The essential point in this definition is the finiteness of the degree in t. The set of all fundamental generators is closed under multiplication in A. Let F(t, &) be any fundamental generator. Then if F (¢, §) is the product in 2A of fundamental generators with coefficients in the domain V, the generator is said to be reducible in V, otherwise #rreducible. Suppose now that each fundamental generator is the product in A of fundamental generators irreducible in V in one way only, apart from permutations of the factors. Then V is called a fundamental domain. In what follows it is assumed that such a V exists. An instance is V= the domain of rational integers for the special and important case in which the generators are polynomials in both & and ¢. It F.I.f, then 1/F-I.f, where .f = qf, the reciprocal in € of f, 5 being the unity in €; if ¥ is fundamental, fis called fundamental, and 5/f the reciprocal of f; both f and n/f are called prime or composite in € according as F' is irreducible or reducible. - Since 5g = g, where g is any element of €, unit factors g are disregarded, and the like is to hold in what follows. A generator G obtained from a finite number of fundamental generators by a finite number of multiplications and divisions (in A) will be called rational; if G-I'-g, then g is called rational in G. The set of all rational derived matrices in € is an abelian group, say Og, under KE multiplication. HF-I.fwe define £7 by F7-T. f7, and clearly F—7.I.f7, where 7* == gif. Let F = eH"... K° be the resolution (in V) of the fundamental generator F into a product of powers of distinct irreducible fundamental generators G, H, - - -, K. Note that f” as just defined is the D product of » derived matrices (in €) each equal to f. The D product of several such powers g% 4°, . .., k° of derived matrices will be indicated by juxtaposition, ¢g® A’... k° so that the last is an element of € and (see § 6) is indeed the £ product of the ath, 124 ALGEBRAIC ARITHMETIC. bth, ..., cth powers g, /,---, kin €. For F,..., K as above, it follows that if : 2-0-5. G.Cig, HDD, +, K-T-.}, then f= gf)... It, wir = nigh. k and further that these resolutions are unique. We shall abbreviate the Z£ product of f and #/g to f/g. A generator F' obtained from a finite number of funda- mental generators by a finite number of multiplications will be called ¢ntegral; it F-I'.f then f is called an integral element in €, and fis prime or composite in V. Hence we see that any integral f in € is the product of prime integral elements in € in one way only (apart from permutations of the factors), also that the reciprocal of the integral element J in € is uniquely the product of reciprocals of prime integral elements in €. Hence any element of € whose generator is in Gg has (apart from powers of the unit 5) a unique resolution of the form HER Ime where g, hy, ---, k, I, my, --., nm are prime elements in €, and the nents are Britian = 0. For convenience we define f° = 4, Whore J is any ele- ment of €. The above shows merely one kind of arithmetic that can be obtained from €. Primes and irreducibility can obviously be defined in many other ways to yield in € the fundamental theorem of arithmetic. The above was se- lected because of its immediate applicability to the codrdi- nation and extension of the algebraic properties of a large body of existing arithmetical functions, as will be briefly indicated in the following sections. THE VARIETY €. OF €, § 10-12 10. Simplifications of § for rational integers. To apply € to functions of rational integers take x, = p (p any prime) in (3.2), and hence x, = = in (3.5) and everywhere APPLICATIONS OF THE ALGEBRAS €, D. 125 in €. This determines a variety, which we shall denote by €,, of €. There is still a wide choice in the definition of Je) = fulx) of (4.4) in §,. The following will suffice for purposes of illustration. Since now x, = p for p prime we take Ju (2p) = Jn (p) (p prime >1, n=20,1, .-) and (see § 4), fo(p) = 1. For this choice (4.4) now defines the function f,(n) = f(n) in E,, Jo) = fa@phB---£0) SO =1, where n = a®f%...r° is as before the resolution of »>1 into primes. In 88 7, 8 we now have zz; = dé (3 any prime > 1), and (7.2) becomes 10-27 gn) = Surin, in which, to obtain the corresponding generator, we replace J by & and ?, by t. The generating identity becomes 76, 5.1.7, Fl, = 1+ 27 6,700), and the specific forms of the f,(§) are still at our disposal. Before specializing we introduce some simplifications appro- priate to €,. As a working device we note first that by the definition of matric equality, which states the equality of derived matrices in €, corres- ponding to those defined in € by (4.6), is equivalent to Fn) == gn) {n=:1,92 -.4. In the last we may drop the reference to the range 1, 2, ... of nm, write simply f(n) = g(n), and understand by this the 126 ALGEBRAIC ARITHMETIC. implied matric equality in €,. Hence the equality f= g/--. I in €, is equivalent to JS (n) = Pp (9(n), h(n), SL k(n), and this in full (on supplying the definition of the D product indicated) is Fm) = 2 gd) hidy) --- k(ds), the DY referring to all sets (dy, ds, ---, ds) of integers dy, dy, +--, ds each > 0 whose product is #, where s = the number of functions g¢, 4, ---, k. The unity in €, is 4, where 5(n) = 1 or 0 according as n=1 or n>1. If fis any element of €,., 4 / =F, the product and the equality having the meanings just explained; the reciprocal f’ of f is uniquely defined by ff’ = 1. To write down the generator of f(n) we have in any given instance the specific forms of the functions Llp), ilpprime=1, 0 = 0,1, +, with f,(p) = 1, and hence from the resolution n» = a*gb ... y¢ we get by the definition of f(n), fn) = f(a) 7,08) -- fly); whence, from the properties of the parameters tg, #, in D we have 1,2" nul =270 700) on observing that t, = ¢, £- . t, and noting the coefficient of t, in the distributed form of the left, so that, replacing (15, 0) by (¢, &), we get the required generator Pitt 30m pin, Conversely, from this generator, reversing the steps just out- lined, we find the f(») generated by F'(¢{, §). We have APPLICATIONS OF THE ALGEBRAS €, ©. 127 insisted on these extremely simple details because the appli- cations of €, involve nothing more complicated. The specialization of the above in which the f, (§) (n=1,2,..-) are polynomials in 5, or in which all but a finite number of them vanish identically and the rest are polynomials, cover practically all of the important arith- metical functions in the literature of unique factorization in rational arithmetic. The following short selection will illustrate operations in €, and prepare the way for the statement of a generalization of €. 11. Certain functions in §,. The most immediate appli- cations of €, are to the extensive literature summarized in Dickson’s History, vol. 1, chapters V, X, XIX, and to func- tions of [z], the greatest integer < x (this section of arithmetic has not been reported in the published parts of the History). These applications concern relations between functions of divisors, the inversion of such relations and of series, gene- ralizations of Euler's ¢ function and others, numerical integrals and derivatives (these are merely a very special case of multiplication and division in €, and have no characteristic properties which distinguish them from other similar operations in €,), and many similar topics, the majority of which €, reduces to a simple, coherent system abstractly identical with the simplest parts of A. We need give only enough to illu- strate the use of generators. The notation is as in § 10. Unless otherwise stated prime shall mean prime => 1, and the functions considered exist only for integral values >0 of their arguments. By nth power is meant the nth power of an integer > 0. If m, n are coprime integers > 0 and f(mn) = f(m) f(n), J is called factorable. The examples in this paragraph are confined to factorable functions. If » is divisible by no square >> 1 it is called semple. Any integer » >> 1 has a unique resolution into a product of coprime simple numbers whose exponents are distinct, n =P? (QP... Bt, where ,Q, ---, B are simple and coprime, and a >b >... >¢. If n = pj... pi is the resolution of » into powers of distinct primes, we call 128 ALGEBRAIC ARITHMETIC. (after Sylvester) A(n) = ay + as + --- + as the multiplicity of n, and y(n) = s the manifoldness of n. Hence i(n), y(n) are respectively the total number, and the number of distinct primes that divide n. A great many of the arithmetical functions in the literature are special cases of the following, or are products in €, of positive or negative integral powers of it: : (11.1) Liln; a,h,c) = 0 if » is not the bth power of a simple number; in the contrary case the value of the function is ¢”’ #n*”. From this definition we write down its generator, (11.2) (Q+e20).T. 000; a,b, and hence that of its reciprocal ¥&, (11.3) (Ate yd. (nn; 0,0 ¢)- The verbal definition of #’ is read off from the expansion 1—c8o tr c28apb — Bab... of the generator, and is (11.4) Win:ia be) = 0 if » is not the bth power of an integer (not necessarily simple); in the contrary case the function has the value (— cm palt, where m = n'*. If s is an integer = 0 we indicate the sth power in E, of & as in (11.1) by writing Win; a, b,c) and so in all like cases. Hence Wn: a,b, 0) = P3(n; a,b,c) From the definition of reciprocals in €, we have A185 Pp al, 0), Fm; a,b, 0) = y(n), since the generators of &s, &—sare the reciprocals (1c &*#)5, (AQ+4tc&®)—= in A. In full (11.5) states that APPLICATIONS OF THE ALGEBRAS €, D. 129 2 Ws (d; a,b, 6) (3; a,b, 0) = gn), the summation extending to all pairs (d, 0) of divisors each =>0 of n such that 4d = nn. The abbreviated form of (11.5) tle Wt == yp, To see that & and its powers in €, contain many of the known arithmetical functions we first specialize as follows, defining six subcases. (11.6) vw; a,b) = Tin; ab —1), = 0 if » is not the bth power of a simple number and otherwise = — un? according as the manifoldness of » is even or odd. (11.7) Yin; a,b) == 0 or n*” according as » is or is not a Oth power. (11.8) 3 (n;0,0) = Win;a, bh), = 0 or » according as » is not or is the Oth power of a simple number. (11.9) 1 (n; 0,0) ==0 if » iz noi a bth power, and other- wise = + n%* according as the multiplicity of »!/* is even or odd. (11.10; a, 8) = ¥(n; 0, b, a), = 0 or & according as n 1s not or is the bth power of a simple number. A115 0,0) = 0 if » is not a bih power, and in the contrary case = —+ atm = n° according as the multiplicity of m is even or odd. The generators are written down from these definitions or at once from (11.2), (11.3): (1—& i) SE v, (1—&a Zt Sh rt, Q1A2y (1-80). 0.3," (LPP. Pai (lta). T.L, (Q-kaly I. 0, the second column being of course superfluous—it is included merely for ease in verifying the special cases given presently. To illustrate in passing one way in which generators are used, we have as the equivalent, in €, of the identity 1—&20 $e — (1— Sa t) (1+ Sa tr) in A the following, Poin; a, 1), x(n; 0, 0) = Win; 24,20), 130 ALGEBRAIC ARITHMETIC. Particularizing these again, and introducing the customary notations for such of the functions as are current, we write p(n) = 0 if n is not simple, otherwise 1 or —1 according as the manifoldness of 2 is even or odd (M¢bius’ function); tam) = p(nt*), which exists only when » is an ath power; k-(n) = 1 or 0 according as » is or is not an rth power; wn) = nn", defined for all real values of =, r, and wy (01?) = wypo(n) and similarly for higher roots than the second; a(n) = 1 or —1 according as the multiplicity of = is even or odd, m(n) = (-1)*®; v(n) = the number of divisors of nn: 9 (n) = the totient of », = the number of integers < n and coprime with 7; 6(n) = the sum of the divisors of =; #(n) = the total number of decompositions of 7 into a pair of coprime factors, the order of the factors being considered. These are but a few of the special cases of & and its powers, From them we get Wn; a, b) = ky (0) pap (n) wep (n), yritln: a,b) = 0) uypln, 2(n; a, 0) == ly (ny {pnp On)l2 wap ln), (11.13) Ams a, b) = (—1)* Joy (0) wap (m) (m= nt), Ln; a, b) = {mpn)}? ky(n) o7®, In; a, b) = (—a)*™ ky (n) (m = nll). Before writing down the generators of special cases of these we must recall the absolute product | fg | of f= (f (1), £(2),---), g= (gl), 9(2), .-+). Puwi'lfy|= hk for the moment. Then (see chapter188), A(n)=gn)gn) n=1,2,...). Hence SE = |ffl|, fF? --- are defined. These will not be confused with the products and powers | fy|, 9° +. in &,, which (as explained above) are products Bo For example 2? |»*| are distinct functions, vin) = 2 v(d) »(9), |72|(n) = {»(0)}?, APPLICATIONS OF THE ALGEBRAS €, 9. 131 the summation referring as before to all pairs of conjugate divisors d, 0 of n. It will suffice to write down the generators of only a few. We see from the above, or directly from the definitions, the following: FUNCTION, GENERATOR FUNCTION, GENERATOR Hn, Ti, lw 7 |, (1—E&H2, |? |, 1-41, Jo), (1+ a—t3, v, (11 ev, 1+ 2¢, Ps {1—7) (1-5, 7h, (1—10 1-1-1, 0. (1 + t) (1 AI i 7T Up |, (1 A 2a, T, a-+71, mv], (14-12 kr, A~vyy lur |, (1—=&72 og, A—aq U—3i |nel, U+070- 5m, Hy, (1-291 lor], +E —-Daa—p, wl yy, wal, 1—E (1— 8g), vin, L241, and the list may be continued indefinitely. This illustrates the richness of ¥, which is itself one of the simplest algebraic generators in €,. If we take V of §9 to be the rational domain, it follows from the foregoing generators that u, |¢?|, nm, |u|, uy, |wu,|, are primes in €,, and that we have the following resolutions into primes in €,; vy = 2, go pu, Gmma, Y = |? uy, |%0| = pn, |up| =u, = |p|} no] = uw), luo = wu, ur] =u, and many more by inspection. For example the resolution of 6 into the product of the primes |u?|, u, is equivalent to the identity in 2[ between generators RO? == (sly and read in full the resolution § = |u?|u, in €, states the relation 6(n) = {1 (d)}?, g* 132 ALGEBRAIC ARITHMETIC. the > referring to all divisors d of n, since u,(d) = 1 for all integers J > 0, ; Having resolved a given set of elements of ©, into their prime factors we may then proceed, precisely as in 2, to obtain from them by operations in €,, of which multipli- cation and division lead to the more interesting results, chains of relations between functions of divisors. We shall illustrate this in a moment. For the present we remark that the equivalent in €, of the associative law of multiplication in 2 leads to interesting consequences. Thus, if £;(j = 1,2, ---, 5) are elements of €,, their product f is in €, and likewise for fufy--- fe, where a, b, ..., ¢ are any integers chosen from 1, 2, ..., s. Each such product defines a function whose verbal definition can be read off from its generator, and the function so defined will in general isolate properties of divisors sufficiently different from those defined by its factors to make it of interest. For example, the product wuy may be replaced wherever it occurs by ¢. Regarding J as a product in all possible ways given by the associative law, thus fifa. fs, fifo-fo Ju: fs ete, we obtain its - structure in €, with reference to 77(s) distinet arithmetical functions, where 77 (s) is the enumerative function of chapter 1, § 26. Again, the most obvious consequences of division in A imply relations in €, which are not always obvious at first sight. Division in A, that is, division of generators in the present instance, may be replaced at once by division in €,, since the processes are abstractly identical, and like- wise for multiplication. For example, suppose fg, I, I, p, q are elements of €, between which there are the relations fg = 2k, Jo = Ly. Then, by elimination of f, 7, precisely as in A, we infer gq = pk. This almost trivial example in €, illustrates the simplicity and power of the method. For, stating the theorem in common notation we have the following: the relations 2f@g0) = Zh@E@ ZF d)p©) = hd) qs) APPLICATIONS OF THE ALGEBRAS €, OD. 133 together imply 29d) q@) = 2pd)k(©), where f, --., q are arbitrary single valued functions of integers, and > refers (as before) to all pairs of conjugate divisors d, 0 of n. As another, similar, example we see that wy f = g¢ implies f = g/u, = gu, since (by their re- spective generators 1—¢, (1—¢)~1 given above) wm, u are reciprocals, so that ww, = 5. But this is the important inversion theorem of Dedekind, namely 70) = gn)... 1 (0) = >gldynld), This theorem (which as a very special instance of division in €,) has therefore the immediate extension to any functions Sg by : : Jo =0L.D.0 =P/f an) f= 7lg or, if FL yg denote as usual the reciprocals of £, g in &, from fg = hh weinler f = hygt, y = Aft exacily as in %, Returning to the special functions we have the following illustrations of the above remarks, UP = Ww, 0, =u, ty ag — "y Uy =r, tly | 1 04 == uu, ut, = u, ao, 70 = pul = 2, md = wl] — wm | = ty 52), pr = pu, "gy = wu, u, = g, 2 =— o" "? = v u,v : aly — wud = yy 7 a ky, wy == Tus = yy k., and so on indefinitely. Again, each of the following, and hence all products and quotients of positive integral powers of any number of them, is equal to #, the unit in G,, 134 ALGEBRAIC ARITHMETIC. pre, ph, wlpwl, mp, pv, |pwu|eu,, | | | | lpm |po, apd, |\mw||mp?|, |pu|u., and so on. To see one of these in full, consider 7x 6H. Then mul = ng states that Dw (dh) ld) lds) = 1 or 0 according as » = 1 or » > 1, the > referring to all triples (di, ds, ds) of divisors of n such that di ds d3 = m. All these are instances of (11. 5). These few examples are sufficient to show one direction in which €, may be used to coordinate and indefinitely extend one part of the algebra of arithmetical functions. We pass on to illustrations of different types. 12. Miscellaneous applications of §,. Again we make only a short selection, as the abstract identity of &, and A suggests inexhaustible possibilities which, however interesting in themselves, add nothing to the comprehensive theorem that €,, A are abstractly identical. We shall endeavor to choose examples that have some Impanioe in other depart- ments of arithmetic. Consider first the arithmetical inversion of products, an instance of which occurs in the determination of the equation of primitive roots. We have defined /“, where fis in €, and a is an integer (=0). By obvious means, precisely as in 2, we can extend the definition to powers whose exponents are any rational numbers. But it is more interesting to generalize in another direction, and we shall define f9, where f, g are any elements of €, to be the product (12.1) 7 = [] rere where [| refers to all pairs (d, 3 of conjugate divisors of x. As before we omit from f? = f9 (n) the symbol of the argument n, and an equality (12.2) fo = Ik, APPLICATIONS OF THE ALGEBRAS (€, 9. 135 where fg, hk are in €,, is to be understood in the matric sense 12 (n) = (wm) n=1,2...). Let 4, k' be the respective reciprocals of ¢, k, so that 99" = kk’ = nq (the unit in €,). Then we have the general imversion of functional powers in €,, expressed in the following (12.2) implies 12.3) FE == 7, Observe first that if for the moment f. g, /, & be interpreted a3 elements of A, so that 4 = 1/y, I = 1/k, the theorem is true. Hence the inversion in €, is abstractly identical with the extraction of roots in 2—the arithmetical roots being understood in 2A. The inversion in €, is proved imme- diately by taking logarithms in (12.2) and multiplying in ©, the result by the arbitrary element ¢ of ©,, vg = 9k H (log f(n) = F(n), log hin) =— Hn)); whence f79 = }%, and therefore, on chosing 9» = ¢'k’ we get (12.3), If in the above we take g = wy, k = 5 we see that (12.4) f= 1.0. £1 = he, or, in non-symbolic form, 241) [1r@ = we) 2. fo) = [[ @*@ a theorem of Dedekind, first used in the theory of binomial congruences by Gauss and Cauchy. “Consider next the inversion of series, and for brevity assume that all the series discussed are infinite and convergent. Write (12.5) Pis=1 (2) (47). i=1 136 ALGEBRAIC ARITHMETIC. Let 2" be the reciprocal in €, of /; multiply (12.5) throughout by #'(j) and take the sum with respect to j — 1, 2, .... Then we see that (12.5) implies (12.6) 14) = TF). = The pair (12.5), (12.6) may be considered as inversions of each other. A special case of the last is of interest. We note particularly that it is a specialization and not, in spite of its appearance, a generalization of the preceding. With Cesaro (see Dick- son’s History, loc cit. for references) let & (x) = x, and let (2), e5(x), --- (&, B. ... = 1, 2, ...) denote single valued functions of x such that, if «, 8 are any intergers 0, Eq (¢5 (z)) = €up (x) = €3 {z, {)), and let 7; g, i, - - - be any elements of €, (= functions taking single definite values for integral values — 0 of their arguments). Then we may write, as in developing €, Fh) = alo) == 1.00) and consider the first of these as the particular element 7, (n) of €.. Thus, defining g.,(n), 2iz(n), - - - similarly, we see that Jus Yas hx, - - - are instances of the arbitrary £, ¢, 4, --- in €.. Hence if between such f, g, h, - -- we have established a relation R in €, an instance R' of R is obtained on supplying to each function the suffix x with the above meaning, and conversely from R' may be inferred R if we take é,(x) — nx and in the result set x = 1, as clearly is permissible by the definition of the &s. Hence R, R' are formally equivalent, and B can be written down from R' by dropping the suffiz x. Now let Q4(x) = 1 or 0 according as the integer x >0 is or is not a member of the class 4, and consider only classes 4 such that QL (0) QA (y) = LA (ay), APPLICATIONS OF THE ALGEBRAS €, D. 137 and hence £4(1) = 1. Then, the meaning of the absolute product | fg |, for any elements f, y of €, being as before (see § 11), we have Pen) — Bow), lea) — Soll and therefore | 27 f, is defined. It follows that in any relation R between arbitrary elements f, g, hy, --- of €,, we may obtain «a formally equivalent relation Ron replacing these by A LOB z Oe fel 24 0 |; Sn respectively, where A, B, C, -.. are any classes of integers, Ty Y, 2, ++ any variables, and £, (x), e5(y), &,(2), --- (et, 8, 7, --- — 1, 2, --.) any sels of functions having the group property as defined for e.(x). It is to be noted that the variables are mot necessarily independent, nor the eq(xz), e3(y), --- necessarily distinct, and that 4, B, C, -.. are not necessarily mutually exclusive. To pass from R' back to R take &q(x) =a e3(y), =By,---(e, =1,2,..), A=B=C=... = the class of all integers > 0, and in the result put = y =z =... = 1, Noting that the reciprocal in €, of 24 (n) 7 (n) is 4 (n) 1 (n), where 7’ is the reciprocal of 7, we see that the inversion of (12.5) into (12.6) implies and is implied by the following, x 42.7) F.(1) = 2 04000) 1. 6) d= tmplies (12.8) Fel) = 2 QA) I (5) Feld). = If in this we replace the functions with suffix x by their full definitions, : Fl) = Fi), J=8) = Fle), we have Cesaro’s general inversion which thus appears as a special case of (12.6). 138 ALGEBRAIC ARITHMETIC. We shall next indicate how €, €, combined with an in- genious use of infinite series due to Hermite (Acta Muthe- matica, vol. 5, pp. 297 —330; Jowrnal fiir Mathematik, vol. 100, pp. 51—65, and several letters in the Correspondence with Stieltjes), may be applied to functions of the greatest integer [x] < xz. By the usual convention [z] = 0 when z is nega- tive. Although it is not necessary in what follows to use infinite series we shall do so for the reasons stated in the Introduction, also because some of the most interesting formulas in the literature of [x] (those relating to the class number,) were first found thus by Hermite, and his method is by no means yet exhausted. A reference to his use of series will show that they suggest transformations which would be unlikely to arise from elementary methods. Hermite ignored all questions of convergence in his memoirs; the justification (if any be needed) of this procedure is contained in € as developed in Chapter I. All series are to be con- sidered as the ( associated functions of the matrices of their coefficients. Hermite’s fundamental remark is that if Ll) == > Ftlmyar, then (12.9) Fl) /(1—x) = oe LO) FFD --- FF (n/ al) 2, which is obvious on expanding (1—x)! and collecting the coefficient of 2”. We shall write | f(n) for the summatory Junction of f(n) taken between the limits 1, mn, that is Jr = Sra), and as before, if f. g are in €,, fy is their £ product, so that fon) = 2 £(d) g(0) extended over all pairs d, d of conjugate divisors of n. Hence in particular wn fim) = 2 10h), APPLICATIONS OF THE ALGEBRAS €, 9. 139 Wire ur(n) = Ww", as already defined in § 11, and hence if (12.10) F(x) /(1—x) =3 [ruin Taking f = wu, in the last we have Hermite’s generating identity for [n/a], : : 1 (12.11) . — np n/a). i—% 1—o — Again, at once from the definitions xR = 12.12) 9%) F'n) = Sarg sia, =1 and therefore (12.15) _ g(a) Tilt) = — 2 [afm). La=1 n=—1 e Multiply (12.10) throughout by g(a), sum for a = 1, -.., oo, and compare with (12.13). Then, since gf — fg we have {12.14) > g (a) (tn a) = = fa) [4 (In/al), of which there is the useful special case obtained by taking g = , (12.15) > [wrap — we =! n/a) f(a). = The last was first given by Liouville and Dirichlet. In the derivation of (12.14) we get incidentally 1216) [rom = 2 g(a) [ranian, whence we have the useful known theorem (12.17) [ror == 3 (nia) flo) == > [ruin [12/al) 140 ALGEBRAIC ARITHMETIC. From the inversion (12.6) we see that if Hm) = 2D hina) g(a), a=1 then oR bly = hd 1 (n) H (n), n=1 where gg, = 5, and hence in particular if either %(n) or H (n) vanishes when » > N, N All) == 2 9 (n) H (nm). n=— Applying this to (12.16) we replace n therein by [x/n], take Hn) = [or rm, CR) = hin), [p/n] = N, and get as the inversion of (12.16), (12.18) [ra = 2 (a) g.f ((n/ al) (99: = 0), on replacing [x] by n. Hence the inversion of (12.17) is (12.19) [rm fe ~ wa) J op tay, since wu, = 5. Note that (12.18) is a pair of inversions, since f, #7, gu may be replaced by g, f, fi respectively. We shall next give enough examples to show how the formulas (12.14)-(12.19) are applied in connection with factorizations in €, as in § 11. In the derivation of (12.14) (12.19), all of which can be proved almost by in- spection, we made only a slight use of Hermite's device, but enough has been given to suggest its utility, for example when applied to the series for theta quotients. The mul- tiplication by the series for (1—x)~! is equivalent to a summation (arithmetical integration); the change of x to 2“ in the function to be thus integrated introduces the greatest integer function. APPLICATIONS OF THE ALGEBRAS €, D. 141 For variety we shall give some examples concerning elements of €, other than those already used in §§ 11-12. Let o,(n) = the sum of the rth powers of all divisors of n, 0 (n) = o(n), 0,(n) = v(n); ¢r(n) = the rth Jordan totient of n, ¢;(n) — ¢ (un), and ¢,(n) = the number of integers 1, and in every other case the value of the function is 1; m= 1.80 n=vn, r>0, and n,=1, or. u (n,)=1, 7,(n)=—1 i n=nln,, r>0, and p(n,) =—1, 7+(m) = 0 in all other cases, so that yu, 7» = g (= the unity in €,). and yo» = nn. The reciprocal of ¢, is ¥, defined by : 1 / 1 : ; yr(n) =n (1— 7] ls (1—); tin) = (1—p)... 1—¢), Pp q where p, --., g are all the different prime divisors of n. In the following examples we shall assume such resolutions in G, as are necessary; all are found immediately from the generators as in § 11. Thus, for example, u,, wu are reciprocals, so that opt = 4; also wu, = g¢,, ete. To simplify the printing we omit all limits from the summation signs, understanding always that > refers to @, and continues so long as all arguments of functions in the summands are integers 0; n in the examples is an arbitrary constant integer 0. Under the | sign the implied sum is with respect to n=1,2, ..., n, as above. From (12.17) and its inverse (12.19) we get, among many others for the functions defined, pairs of inverse relations as follows. Take f = u,; then uw, f = ul =», Su, m) = u,; hence I (n) = > [n/al, i = > ua) [v(nla). 142 ALGEBRAIC ARITHMETIC. Similarly the choice f = ¢, gives the pair Jw = Zwlage) = fo. (ian, [or n) =A" + 2+ -.. + [wlal) pula); taking f = mn we have wuyf = k;, and hence the pair of inverses [07] = XY walm(a) = 1 oh [= (0) == ~ [(n/a)?] pu (ar). Let S,. s(n) = the sum of the sth powers of all divisors of n that are rth powers. Then the choice f(n) = n° k, (n) gives [ 8, +(n) = 2 nlalat ky (a) = 2X [n/a] a7, which, for s = 1 is a formula due to Lipschitz, [9 (7) == > [n/a] a”. The inverses are uninteresting. Combining the results of taking f= n, f= yr, F = kk successively we find the curious result Nn (@ fn (wal) = Du OIE (w/al) = 2 | 4 ([n?/al). These may be continued indefinitely. From (12.16) and its inverse (12.18) we write down the following examples. Take g = ¥,, f = w,. Then gf = uy, and the reciprocal of w, is |u,u|, that of v,. is ¢,. Hence n= +2+ -.. +a), (a), and this gives the pair of inverses [or = M[n/aly, (a), | i: (n) == [n/a] ar wa). The choice f= uy, gy = |#?| and the factorization | u?| uy = 0 in €, give the following, due to Dirichlet, APPLICATIONS OF THE ALGEBRAS €, D. 148 fo 0) = 2 p*(a) [n/a], and hence, since |u%| 7 = », we have the inverses s [4° | Ns n=) | 8@/a) = Zo) [ (usa), [ur = 2 ula) I (n/a). Taking f= [ui hy|, g = 11 p0r| We get nn+1)/2 = 21+ 2+... + [/)VT} ap, (a), where the absence of » on the left is noticeable, and this gives the inverse (we omit one) 2) appa) == ~ (n/a) (14 n/a) a w(a). The choice f==w,, y = | p| gives fg =, and hence 2 = n/a A+ [n/al) apa) which is its own inverses. By a slight variation of the extremely simple technique we write down the following in conclusion: nnt+1) = 2 De (n/a)) + [n/a] gp(a—1)}, 1 = 2X {uln/a)+In/dlpla—1)}, . [2] = 2X {a(n/a)) + n/a) nla—1)}, o fe (0) == N fn/al {{n/al—14+ (a—1)}, 1 = X' Pinte, where P(r) = the number of primes 1, satisfy WW == Hg Nag -- » Ny Hes 1, 0,7), and let L = Jj (a1, su) (5 = ie: en t) be ¢ numerical functions. Then the #-fold D product fi fs --- ft with the matric argument N, = (ny, ns, - ++, m,) is defined as : : HL >| | 75, me. + wn. Ji As in €,. we may omit reference to N,. The unity of this multiplication is easily seen to be the numerical function ow == wim, «Zr defined by the properties w = 1 when x; = 1(j = 1, ..., r), uw = 0 otherwise. Evidently uf = f, where fis any numerical function (of » arguments). The zero of multiplication is the function «wv which vanishes for all matric arguments, wf = w. Precisely as in € or ©, it can be shown that / F w has a unique reciprocal f’, that is, the equation ff’ = wu has APPLICATIONS OF THE ALGEBRAS €, DD. 14H one and only one solution (numerical function of » arguments) f”. Hence multiplication has a unique inverse and is abstractly identical with multiplication in 2A. Let C be the numerical function which takes the value 1 for all values of its matric argument N,, and let {' = wu. Then it follows easily that {'(n,, -+-, ny) = 0 if the n;(j = 1, ..., 1) are not all simple, and in the contrary case the value is 1 or —1 according as an even or an odd number of the n; have odd manifoldness (see § 11). Hence each of §, { is a solution = of v(, Na, » ++, By) = v(m) v0) - = - 2{n0,Y, and we see that if /, ¢ are any numerical functions (of » arguments), g = tr0.f=1y, which is a generalization of Dedekind’s inversion (» = 1). Again, precisely as in €,, fu = hy = Uf and of = be, af == by ond ah = 001.20. fk = 4h, and so on. The abstract identity with ¥ gives, as in &, an infinity of such relations. The functional powers f7 also exist as in G,. There is a theory of associated functions here as in 9, but not one of generators—as in €,. Finally, as already mentioned, the extension can be readily made to functions of » arguments belonging to » distinct semigroups. 14. Other instances of €. From the manner in which € was constructed it is clear that the integer 2 in (3.4) may be replaced by the particular general element »n of any variety 8 in which the fundamental theorem of arithmetic holds, provided «, 8, -.-, y be replaced by the prime factors of n in B. For instance, n, a, 8, ---, y may denote ideals in a given algebraic number field, n being considered principal (corresponding to a given algebraic integer) when «, 8, ---. 7 10 146 ALGEBRAIC ARITHMETIC, become the distinct prime ideal factors (not necessarily principal) of n. In any instance the uniform functions f, (7). --- on which € is constructed can be chosen only such that addition and subtraction are significant for them. Thus, in the case of ideals we might choose for f,(z,) the norm of «*. The further development in any instance proceeds precisely as in €. It is immaterial according to what law the elements xz, (n in PB) are arranged into derived matrices, provided the first element be that corresponding to the unity in 8, as we work only with generators and the general elements of the matrices. It is sufficient to know that the x, for 8 can be placed in (1,1) correspondence with those for €. The last follows since the elements of ¥ are denumerable, by our assumptions in Chapter I concerning arithmetic. APPLICATIONS OF € TO THE ALGEBRA OF SEQUENCES, §§ 15-17 15. Blissard’s umbrae. The applications in question are made by means of. the variety B of € described in Chapter I § 22. As the subject is very extensive, and its applications numerous, we can give only the briefest outline. A systematic use of the algebra sketched here greatly abridges the computations necessary in the algebra of sequences of functions or numbers and, what is more significant, suggests many interesting extensions or generalizations of well known theories—for example those of the several kinds of Bernoullian functions in existence (including their usual generalizations to functions of several variables), the like for the Eulerian functions, and the essential generalizations of all these which result when the numbers of Bernoulli and Euler are replaced by the polynomials occurring as coefficients in the power series expansions of elliptic functions. The algebra has in fact immediate and fruitful application to any functions containing in their definition an arbitrary integer. The consequences of representing the entire class of elements cn (m=0,1,...) of A by the single letter or umbra ¢ were first developed by Blissard in a series of papers on the Theory APPLICATIONS OF THE ALGEBRAS €, D. 147 of Generic Equations in vols. 4-6 of the Quarterly Journal. issentially the same theory, with less detail, was stated about 15 years later by Lucas, who also gave a short account of it in chapters XXIII of his 7%éorie des Nombres (1891). The theory of Blissard’s method can be carried much beyond its current state; it can indeed be developed in complete isomorphism with 2. The extension of the method by the introduction of umbral division, umbral differentiation and integration, the umbral circular functions, ete., lead to par- ticularly interesting consequences. As Blissard’s work seems to have been overlooked by many who attribute its simple and ingenious processes to later writers, I have designated the algebra based on it by B to recall the name of its originator. In a certain sense B includes A. This will appear so far as multiplication is concerned presently; I shall not take the space to consider true umbral addition and division (not discussed by Blissard or Lucas), which com- pletes the: inclusion... 1 (gy, ¢,---) i8 any matrix in ¥ we shall denote this matrix by ¢, and call ¢ the wmbra of the matrix; c¢ is not an element of 2A. The interpretation of the umbral power ¢ is that this power represents the nth element cn of the matrix (co, ¢; - ++) of which c¢ is the umbra; ¢° repre- sents co. If c is any umbra we shall write *=¢, =0,1, --.). In dealing with several umbrae it is occasionally necessary to give them suffixes. Thus, for example, the umbra of the C matrix (bo, byt, - ++, brn, ---) in A may be designated by b,-and we have b= 0b, n==0,1,.:+}, Umbrae are combined in € or B according to the following fundamental rules. Umbral powers ¢, ¢™, ... (m, n integers —> 0) occurring in computations are manipulated as if they were scalars (— ele- ments of A) with the three following restrictions: (1) zeroth powers a’, 0° -.. of umbrae a, b, --- must always be indicated explicitly and are not to be replaced by the unity 1 of A; (2) if in any linear function of umbrae with raised suffixes a given umbra occurs precisely s times, it is to be replaced by s distinct umbrae until after the completion of all opera- tions as in 2A, when (3) all exponents (raised suffixes) of 10% 148 ALGEBRAIC ARITHMETIC. umbrae are degraded to suffixes. Umbrae are equal only when the matrices of which they are the umbrae are equal. To the absolute product |a'b| of a=0,a04d0;:" "3 9=1{00,0- where a”, b, (n= 0,1, .--) are scalars, we assign the umbra a’). Hence a'b is the umbra of (b,, aby, a2 bs, ---), since a® is a scalar and hence — 1 (the unity in A). 16. Umbral T. The I to which we refer is that of Chapter II, 3 7. Let a, U, -+-, ¢c be kk umbrae, 4, %, +++, 2 scalars, and » =~ O an integer. Then by definition n! : : a om te CB. col (wath trop = 2 gr aff ay bye ey, where D refers to all sets of k integers each = 0 whose sum is ». Hence (see chapter I § 22), if ¢ is also scalar, exp xat exp ybt ... expzet = exp (za-t+yb-+-..+zo)t. Hence if / as in ¥. is the imaginary unit, the umbral sine and cosine are defined by 2isin at = expiat— exp (—iat), 2cosat = expiat + exp (—iaf); whence ; 0 (—1)» $21 3 (—1) 2n fa rr mn iy at == = 57, sin at > nll) ont, COR at > @n)! (27, and with an obvious meaning for the umbral derivative a d ‘ ——S8in at = { cos at, —— e08 at = —F¢ sin at, da : da also Sin atl == a cos at g COS al = —gp sin al dt ; A dt : : in the last two of which the indicated multiplication by a is to be performed (see § 15) before exponents are degraded; in the first pair of derivatives the differentiations are per- APPLICATIONS OF THE ALGEBRAS €, D. 149 formed upon the series for sin af, cos af in the forms with raised suffixes. If in the above umbral (za-+yb-..-+ zc)" we interpret a, b, ---, ¢ as scalars, the expansion becomes the multinomial theorem in A, and similarly, mutatis mutandis, for the umbral exponentials and circular functions. We shall refer to this interpretation as the scalar instance of the umbral functions, ‘and similarly for scalar instances of umbral identities. It is clear, that the scalar instance of.an umbral identity Is an identity in 2. Hence any umbral identity can be read in either of two ways, namely as®identity between scalars or as one between umbrae. The second is the more inclusive. For example the umbral expansion of («-+b—a)® can be written FO—a)’+ 2a (b—a)' + a’ (b— a)? = a®0®a®+ 2a (Wa — ba?) + a®(B2a’—20'at 4 00a?) = asby +a —2a30y) + (agbs—2a,by + ashy), which reduces to agb,. Hence (a+b—a)*> = a’? either umbrally or in A, since in A we have a® = For » = 0 an integer we define as the simplest instances of functions to be generalized presently $ula, b) = (a4 by" (a—b)", Wu(a,b) = (a+ by*—(a—b)", where «, h are umbrae, and hence if # is scalar as before 2cosatcos bt = cos(atb)t+cos(a—b)t = cos g(a, b)t, 2cosatsin bt = sin (a+ b)t—sin (a —b)i = siny(a, bt, 2sin atcos bt = sin (a+b)t-+sin (a—b)t = sin g(a, bt, 2sin at sin bt = cos (a—b) t—-cos (a+b) t = —cosy(a,b)t, in abstract identity with T. The first, for example, states that the coefficient of (—1)* ¢**/(2n)! in the product (—1 n 12n (—1 n tn py aan op ~ ry (2 Lika — bay 150 ALGEBRAIC ARITHMETIC. is N20 (a+b) 4 (a—br = 2 3 A (an—2r Dor. yr Since we are operating in € or 8 convergence is irrelevant; the series are merely those associated with the matrices of their coefficients as explained in chapter I. It now follows that, precisely as in obtaining the 9 iso- mo of T in Chapter II §§ 7-9, there is a complete umbral Z, say Z,. In this the most useful formulas are the generaliza- tions of the above to any number of umbral arguments and the abstractly identical equivalents in ZI, of the addition theorems in %. The formulas for composition and decomposition in I are the scalar instances of an important set of formulas in T ~Uy whose application will be pointed out after we have indicated their nature. Let the w;, up, uw), uf, + (= 1,2, -..) denote umbrae, so that, for example, wr =u, n=021. ad in the scalar instance, «; = u!, where u, is interpreted as an element of A. From these umbrae re can now form D matrices, precisely as in the scalar instance. Write Ur = (wm, us, -»-, uy), Vs = (v1, va, -» +, Us), Uy == (ul, wh, ~~ Uy), = = 0, v2, - +, Ug), and as in previous chapters indicate conjunction by plus signs, thus U, + V,. The particular order chosen as normal for the uj, vx in the conjoint is immaterial in what follows (as will appear presently); for definiteness that order may be taken as normal in which the v's follow the «’s, and accented letters follow unaccented. Thus U+V; Ea (211, Uo, 20s Uns Ut, Un, tics, Us); Ur t-Us== (in, tn, --; Wr; UL, Wasi» +, Ws Bach of g, 2; (=1,2,...) as in B denotes a definite one of 1, —1, and XY refers to all possible sets of values of the e's and &’s occurring in the arguments of the summand; the functions ¢, ¥, x are then defined for n — 0, 1, --- by APPLICATIONS OF THE ALGEBRAS €, D. 151 vulUp) = 2 1+ es ust es us + + - - + erm)”, P, (Vo) = eres esr ez ont-- + eva, llr Va) BEX e100 ++ 20m Fatt esi epuy tanta... sol Hence, ¢ being scalar, we have from chapter II § 10, r=] I cos u;t = cos 9 (U,)t, J=i 2s 9251 (—1)7 11 Sin Uj ¢ = COS YP (Vas) t, i=l 254-1 228 (—1)8 11 sin vy; t = sin @(Vysi1)d, = : 7 2s : 2728-1 (—1)8 11 cos uj t- [1 sin ut = cos 200, Volt, Jj=1 k=1 7 254-1 27125 (—1)8 1 Cos w; 1. I Sin vp t= sin x{U,; Vocri)i. j=1 L=1 The gu, Vy, yn are therefore instances of parity functions, and the respective parities of 9a, (Uy), Won (Vas), Wont1 (Vast), 220 (Ur; Vas), domt1 (Uy; Vosta) are p(Uy |), p(Vas |), p(|Vest), Ty, Vas 1), p (Ur | Vos ia). If now we multiply together corresponding members of any pair of such identities and use the above to reduce the results to the same respective forms, we find six addition theorems for the ¢, ¥, yx functions, of which we shall write only two, $20 (Ur + Up) = 92 (9 (Uy), 9 (Up), 22041 (Uy + Up; Vos + Vig i1) = Wau11(1 (Uy; Vag), 2(Up; Vain). Similarly, if each umbra in U, be replaced by the sum of two others, for example vw = vw, so that u® = (v-+ w)® (mn = 0, 1, ...), we find in the same way a second kind of addition theorem, of which a simple instance is 152 ALGEBRAIC ARITHMETIC. 20: Ur, % +0") = gu, {Up WW, 0) 30, Up; w, vu”), where each of «', »” is an umbra (not a matrix of umbrae). Combining the two kinds we get a third, of which a specimen is x20 (10; Vog+1, v) === Pay (ut, WY (Vosia, v)), where wu, v are umbrae. As in I there are subtraction theorems, obtained in an obvious way similarly to the above, and by combinations of the resulting formulas with the addition theorems we reach three types of expansion and decompo- sition formulas in I each abstractly identical with the single set in I. The simplest application of such formulas is to the functional generalization of recurrence relations for sequences of numbers or functions. Let f(#) be a function in A of the type J) = cteattettt..., where the series may terminate. If in the instances 3; (j =r, ¢) of A the series does not terminate it is usually assumed that it is convergent for the values of # considered; if the series is divergent it is interpreted as the associated function of (Coy C15 +++). Then, the > referring as before to the ¢'s, and % being an arbitrary scalar, we have Fete) = 2 f+ A + eytty +--+ ery), with analogous formulas (involving products of ¢'s or &'s as coefficients on the right) for v, yx. Let each of z, 4, pu, .--, » denote a definite one of ¢, ¥, x for given arguments Ug, Vi, ---. Referring to the original definitions of ¢,, (U}), Wu (Vs), 40 (Ur; Ve) we write now Ty (U,) Uy + ets +--+ erry, Toy (Vy) tT &avst gus Tolle: Wo) == Toll oath he opto sbonscko tote APPLICATIONS OF THE ALGEBRAS €. D. 153 or understanding in any case U,, Vi, simply T¢ for the corresponding sum on the right, and Sel) =1, Se¢(V)) eres reer, S(Up; Ve) = 8180-08, with similar abbreviations to S¢. If several T¢, S¢ occur in one formula the e's and &s are to be treated as umbrae; that is, the sets of units pertaining to different functions are to be denoted by distinct sets of letters, thus ej, ¢j, , &, - - - (j = 1,2,...). This is merely to prevent confusion in assigning to the argument of > in the following all possible sets of values of the e's and &s (each = +1). Suppose now that between v, 4, pu, ...,» for the given arguments there is the relation ¢ = A+ p+ ... +». Then, f being as above, we have Jt) = 50) 80) --- 8) f+ TUL Tle) - -- + Tl), as is easily seen. This is the prototype of all recurrence formulas for sequences of elements of either type toy Ua, Ug, He, ~~, Cty Usy Un Ugg nines By the proper change in notation any sequence can be written in either of these forms (with all even or all odd suffixes). 1f the sequences are given directly in the form wy, wy, we, wy, --- the exponential theorem (umbral) is used from the beginning, instead of the circular functions, and we reach correspondingly simpler formulas. Thus if «, 8,7, ---,d are umbrae such that « = 84-74 ... 4 J, then immediately Jiutey= s+ B-ly-1i... 1d). It remains to indicate briefly how umbral equalities such as r=A+4p-+ ... or « =g8-+4 y+ ... above may be found, and it will be sufficient to take as an illustration the simplest example from functions of a single variable. Let f(¢) be an even function of #, and ¢(f/) an odd function in F., both possessing MacLaurin expansions for some [¢/. Then the expansions may be written 154 ALGEBRAIC ARITHMETIC. Flr) == cosa, gD) = sin bi, where a, / are umbrae. If now we have several such fune- tions and their expansions, LD). = coset, Li) = sindi, --- their products define further even or odd functions, and we have for example 210 (t) 1) = coset == cosyln, Ot 20) k(t) = m(t) = — cosrt = — cos (bh, d)t, where we have regarded the products f7, gk as defining new umbrae e, », and we have em == mia, 0), vo == 0,4) m= 0,1,-.), Again, since ff) is even, wy = 0 (n= 0,1, --.) and hence ¢.,+1 (0, ¢) = 0. Thus if we define the e's of odd ranks, namely exi1, bY esni1 = 911 (a, ©), we have e, = ¢, (a, © (n=20,1,...), and hence ¢ = ¢. Similarly for products of several factors. A simple example is given by the Euler numbers ln = 0,1, 2), Boy = 0, defined by sect = cos Li. Hence 2 = 2sectcosi = 2 cos licost — cosopll, 1)¢, and therefore ¢(Z, 1) =u, where wy = 2, uw, = 0(n 0). In general the expansions of any set of functions define a unique set of umbrae; by means of the algebraic or analytic properties of the functions, or by merely considering the expansions of their reciprocals, powers, ete., we generate new umbrae and obtain equalities between the new umbrae and the old. Extremely simple equalities such as these frequently appear as the origin or the core of elaborate theories of the algebraic aspects of sequences of functions or numbers, for example those of Bernoulli, Euler, Genocchi, Lucas, and the Legendre and Bessel functions. APPLICATIONS OF THE ALGEBRAS €, 9. 15H The umbral differential calculus can be readily developed from the following definition, abstractly identical with that for the rth derivative of 4” with respect to « in Fe. ap n! Ti (ly = (lp—r {(n—r)! where 1); denotes the rth umbral derivative with respect to the umbra ¢. We have also, writing D, — d/da, d.. d. : 7 fla-t10) == lets precisely as in FF. The last is a useful formula. 17. Umbral fields. An infinity of fields, regular or irregular, can be devised for the study of special sequences. The interpretations assigned to the four fundamental operations are determined by the sequences concerned and particular relations between them which it is desired to investigate; the flexibility of the method allows it to be bent to any al- gebraic end. For example, if it be desired to extend to negative ranks —m»n a sequence of functions or numbers commonly defined only for n > 0, the algebra at once pro- vides the generators of the extended sequence in such forms that they are consistent with the initially given functions or numbers. The simplest umbral fields are isomorphic to the umbral exponential functions. Two examples will be suf- . ficient. Latin letters a, b, ..., ¢ denote umbrae, Greek a, A. =v sealars. We have defined «a to be the umbra of (ay, «a, «as, - a” (ay, ---). We designate the umbra of (ex ay, cay, ---, ay, ---) by the new symbol («¢.a), = (a-«). From the set of all (ea) generated as « runs through all elements of 2 and « through all umbrae of matrices in 2, we shall construct an irregular field JU. The sum in JU of any two elements (e¢-a), (8-0) of FU will be written («-a)- + (8-0), and their product («-a). (3-0), both of which are to be so defined that they satisty the postulates of an irregular field 156 : ALGEBRAIC ARITHMETIC. as stated in Chapter I § 3. This may be done in several ways. The following is given in some detail as it is typical of all cases. We first, as a preliminary survey to determine the interpretations of addition and multiplication, transpose additive and multiplicative properties of the functions dis- cussed (here umbral exponentials) into terms of umbrae alone. Having seen what interpretations are sufficient we then suppress all mention of the exponentials and construct the field ab initio, verifying independently from the definitions given by the survey that these do indeed satisfy the re- quired postulates. From the definition of («-«) as the umbra of (cay, ---, aay, ---), it follows that exp (a -na)t = Mu inti )ile aft = aa, Now exp («-a)t is in 2, being an associated function of the matrix whose umbra is («.a). Hence, if g(a, 8, ---, 7) is a function in 2A, so also is g(exp («-a)t, exp (3-D)t, ---, exp (y-c)i). From the definition of («.a) we have expat = exp(e-a)i, (1.4) == gq. Take now ¢(«, 8) = «+ 8. Then it is suggested thus that we define the sum in JU of («- a), (8-0) through cexpat-+Bexpbt = exp(e-a)-+ -(8-D)¢, which states that (¢.a).-+.(8.0) in JU is the umbra of the matrix whose nth element is a «,, + 08, in A, and hence {e-a).+.(8-D = an, 80, Similarly, taking ¢ (a, 8) = «- 8 in A, we may define multi- plication in JU through eexpat-Bexpbt = exp ((e-a)-(8-b)i, APPLICATIONS OF THE ALGEBRAS €, 9. 15 =] the indicated product on the left being in 2, so that ((¢-a)-(B-D))' = eB(atb) = «f = %) (nj bj. J=0 Thus we may take as the product (e«-a)-(8-0) in JU of (ec. a), (8-0) the umbra of the matrix whose nth element is the element eB (a 4b)" of A. In the same way the unity (1.4) = wu and the zero (1-2) =z in JU are indicated at once: it is sufficient to take uw, — 1 (the wmity in HW), w, = 00 >0), and 2, =0n=0,1...}. Further («-a) is regular if and only if ea, + 0, and if («- a), (8-0) are given, and (e-«) is regular, then there exists a unique element (y-¢) in JU, called the quotient of (8-10) by («-a), such that {(e-a)-(y-¢) = (8:0), for the last is equivalent to n cy > 2 or CT Bb, (n — 0, 1, re Ds J=0 which uniquely determine yc, (n = 0,1, ...) if and only. if ad; 3 0. : Having thus foreseen a possible set of interpretations for the four fundamental operations in JU by means of an isomorphism with umbral exponentials, we can discard the latter and construct JU independently. Formal proofs that the following interpretations are self consistent and an instance of the postulate system for an irregular field are superfluous, as they are implied by the abstract identity with exponentials which we are discarding. It will however be of interest to show how one or two are proved directly from the postulates. We take as elements of 31 the (e- a), (8-0), ---, G-0), . .. as first defined (without reference to exponentials), so that (0-0) = aun, (n= 0,1,.-.) and (n. a) = (3-0), by the definition of umbral equality, only when («-a)* = (8.0)" (mn =0,1,...). Then (e-a)--+-.(B-b) is defined as the umbra of the matrix whose nth element is «a, + 8b,, and («- a)- (8-0) as the umbra of the matrix whose nth element is «f(a + by"; the zero, unity in YU are (1-u), (1.2) respectively, 158 ALGEBRAIC ARITHMETIC. as above. With these definitions it is easily shown from first principles that JU is an instance of J& in chapter I § 3. For example, the unique negative (1.2). —.(«-a) of (e- a) is (—ea-a), the last being the umbra of the matrix whose nth element is —eaq,. Again, if the associative law of multiplication holds, we should have (aa) ((B-1)-G- 0) = ((@-a)- (BH) (7-0), and hence la-a)- (By bc) == (ef.-ad) -G-0), that is, («By - abc) = (eBy- abc), which completes the veri- fication. The distributive law states that (7-0) a-a)-+-(B-B) = Ga-cay. +. G8 cl). Set (¢-a)-+ (8-0) = (1-d), as obviously is permissible. Then it must be shown that lr-0)- 0-0) = Go-ca)-+-GB: ch), and therefore [e+ (ex -a)-+-(B-D)] = «(c+ a)y*+ B(c+ Db)". The left of this is n n 3 (") en—j (@ a; + BY), Jj=01\/ which is identical with the right, thus verifying the distri- butive law. From JU further irregular fields may be generated. Thus, if 0 is a variable in 2, then 4.0, )=00+-a)y =a 1 neg 0: Jj=0\J defines a matrix of polynomials in A whose umbra is AO, («¢-a)). Call («- a) the index of A(H, («-a)), and write le. 0): L- 1. = 0-9, (eo 0:-00.-1)=(1.-ph Then there exists an irregular field 3114 whose elements are the 4(0, («¢-a)), where («-a) runs through all elements of APPLICATIONS OF THE ALGEBRAS €, D. 159 JU, and in which the sum of any two elements A (6, («- a), (0, (8-1) is A(s, (1, s)), and their product is A(8, (1, p)). Evidently J; L:00,(e.0)) = ndpa(d, (0.0). The further development belongs to the detailed applications of the subject, with which we are not concerned here. From the foregoing examples it is clear that any set of functions in a given variety generate, through the umbrae of their coefficients, irregular fields from which further irregular fields can be constructed by setting up correspondences with given functions. In the second example above the correspondence is with one of the simplest functions in 2, namely (6 + 7)", where 6, y are in A. CHAPTER V ARITHMETICAL STRUCTURE NATURE OF GENERAL ARITHMETIC, §§1-2 1. Abstraction, doctrinal function, transformation by formal equivalence. In the preceding chapters we have seen fragments of arithmetical structure emerging from theories not primarily concerned with arithmetic although applicable to it. The question thus arises as to what extent any theory is arithmetic. In any instance the answer must at present be more or less indefinite if, as suggested in Chapter I, there exists no agreement concerning what can legitimately be called arithmetic. One kind of generalization would draw no sharp distinction between algebra, arithmetic and analysis. From the point of view adopted here this is a misconception; the whole subject becomes entirely too simple and too structureless to be of interest unless it preserves the central features of rational arithmetic 9, namely the existence of integral elements and their unique factorization. What follows is merely a series of proposals toward an arithmetical theory of the structure of any theory constructed in accordance with the accepted processes of logic; it is an attempt to suggest a reason for the constant recurrence in theories applicable to the theory of numbers of an abstract structure which is itself arithmetical. It is complete in no detail. Whatever interest it may have lies in its attempt to show the need for more extensive postulational formulations of the several divisions of rational arithmetic than have been attempted, and in its indication of a new type of arithmetization which perhaps goes deeper than any yet proposed. With the injection of Dickson arithmetics into the theory of numbers it is apparent that “general arithmetic” in the classic sense of Dedekind and Kronecker is too restricted. This indicates 160 ARITHMETICAL STRUCTURE. 161 one direction in which generalizations of the theory of numbers may be sought. Another, to be sketched here, is less of a generalization than an abstraction. Its applications will be chiefly to the comparison of existing generalizations and to the exhibition of such abstract identities of form as are concealed beneath their various interpretations. The detailed application of the suggested program to even the simpler theories of arithmetic is a difficult matter, and is beyond the scope of the present chapter, although considerable progress has been made by my students. The project is feasible, but needs patience for its complete elaboration. The point of view will be obvious to anyone acquainted with the General Analysis of E. H. Moore, as developed in his Colloquium Lectures, 1910, and in his lectures at the University of Chicago, or with the concept of doctrinal Junctions as stated by C. J. Keyser (Concerning multiple inter- pretations of postulate systems, etc., Journal of Psychology and Scientific Method, New York, vol. 9 no. 10, 1913) or again with the abstractly identical concept of the system functions of H. M. Sheffer (7%e General Theory of Notational Relativity, mimeographed, Cambridge, Mass. 1921). For an accessible account of the doctrinal function (we shall use Keyser’s convenient term) we refer to his Introduction to Mathematical Philosophy, 1922, Lecture III. It will suffice here to recall that a doctrinal function is the extension to a system of postulates and their logical consequences, of the Whitehead- Russell propositional function. The doctrinal function of a given part of rational arithmetic, say the theory of divi- sibility, is what we shall call the general theory of that part. Interpretations (obtained by assigning to the marks in the doctrinal function specific meanings such that the truth value of the function is + (= true)) are, as before, called mstances of the general theory. Since the introduction of doctrinal functions into mathematics it is improper to call any instance a general theory. This applies in particular to what is sometimes called “general arithmetic”. For each of the several parts of rational arithmetic, such as the theory of the G.C.D., 11 ’ 162 ALGEBRAIC ARITHMETIC. L.C. M., congruences, etc., there is precisely one doctrinal function; the set of all such, from the present point of view, is arithmetic—we may drop the qualification general. A value of the doctrinal function is an instance of arithmetic, but it is not itself arithmetic. Two complete instances of arithmetic have yet to be constructed; of the many partial instances the majority refer to the multiplicative side of arithmetic. It can be decided to what extent any theory 7 is arithmetic by setting up its doctrinal function and comparing the result with arithmetic, and similarly for any generalization of arithmetic. Before proceeding to the sketch we shall recall some of the terms used, although this is not necessary if familiarity with the cited works be presupposed. In each of the postulates of a given set 7 the letters signifying relations are replaced by marks (Boole, E. H. Moore) denoting variable relations, and similarly for the relata, the marks for the latter signifying variable elements. It is an advantage to choose for the marks symbols bearing no resemblance to the originals. Thus a >>0 may be replaced by B(*,7) or R(X, Y), ete. The logical constants in 7 are left unchanged. The result is the doctrinal function C'(7) of 7. By assigning to the marks in C (7) the interpretations which they have in 7 we recover 7. It may be possible to assign another set of interpretations such that the result, C(T,), is a set of true propositions. Each such C(7)) is a value of C (7), The problem for a given 7 is twofold; first, from 7 we are to construct C (7), second, from C(7) we are either to find its values given the elements (relata), or we are to show that it is impossible to assign -such inter- pretations to the relations in C'(7") that, for the given elements, the truth-value of C' (7) is +. The first part of the problem can always be solved readily. To solve the second, the doctrinal functions for the given elements must be constructed. If none of these is C'(7') the problem has no solution. The last is in nature of an existence theorem; it is as impracti- cable as many of the standard processes of the theory of ideals. ARITHMETICAL STRUCTURE. 163 Let C(T), C(6) be distinct doctrinal functions. Then we shall call 7' a generalization of @ only when the first of the following is true and the second false, AT) DS Co), OE) COT; if both implications are true, so that C(7) :=: C(6), then 7, 6 are identical. Instances T;(; = 1, 2,...). of C(T) are called abstractly identical. HH P,Q are propositions such that P :==: ¢ that ig (PDQ) -(Q D P), either of P, Q may replace the other in an implication involving either. The substitution is called transformation by formal equivalence. Its use is to transform given propositions into others more easily recognizable as being abstractly identical with other given propositions. For example, « — b mod m in rational arithmetic may be re- placed by the assertion that «— b is in the class of integral multiples of mm, in which form it is abstractly identical with congruence with respect to an ideal modulus. 2. Moore’s heuristic principle. We have thus far two problems, (A) Given 7, find C(T); (B), given C(T), find 7. There is now the third problem (C), given C(7"), make C(T) categorical and separate C'(7') into all possible parts C; (7) (j = 1,2, --.) such that each is categorical. Each C;(T) is a subtheory of C(7). We have called C(N) where RN is rational arithmetic, arithmetic. From any 7" we are to isolate all G(T) such that CON)... G(T). Leb j= aD, «=x 0 he these... Then the arithmetic of C(T) is given by J=a,b,.-.,c,-... Suppose now that we are given C(T). Then C(7) may be minimized in the categorical sense, that is, from C(7) are to be rejected all those of its elements which are implied by other of its elements. There remains an irreducible residue C’(7') which implies C(7). If the solution of problem (C) is apparently multiform, giving OP (T) (j=1,2,...), then CO (T) =; CH (T), and the solutions are identical. As already remarked, the actual solution of problem (B) Is impracticable according to the method suggested by its 11% 164 ALGEBRAIC ARITHMETIC. existence proof —a by no means rare characteristic of existence proofs in mathematics. In the practical attack on (B) a guide is the heuristic principle of Moore. This principle, now being accessible in the place cited, may be taken for granted; beautiful illustrations of it occur in the earlier papers of Dedekind on algebraic numbers and ideals, but it was first clearly formulated as a principle by Moore. In what follows this principle will be recognized as resident in the G. C.D. of two theories 7, @. The extent to which two theories are unified with respect to their central features may be measured by their G. C.D. The algebra of classes (and hence also of relations) is essentially arithmetical in structure. By structure we mean the form of a doctrinal function, leaving form undefined, (no satisfactory definition has yet been given) although it may be readily apprehendeds For example, «Db has the same form as p Dg, while # :=: b and p Dq have different forms. It might be presumed that it would be a simple exercise to specify form by the logical constants and variables involved in a given C(7). This however is not the case; it is in general easy to destroy any such attempted formulation by pointing out tacit assumptions as to order (one such occurs, for example, in classic postulates for order itself: “a Rb and bRa are distinct”). We shall therefore assume the meaning of structure as known. That rational arithmetic and the algebra of classes should have much in common structurally is not unexpected since both are based on the discrete. It would be interesting to know which, if either, is logically precedent; it is more interesting to take as a hypothesis the assumption that logic and arithmetic are abstractly identical. The link between arithmetic and structure appears in the fundamental dichotomy of logic. It will be necessary there- fore to indicate how the algebra £ of classes (or of relations) may be exhibited as an instance of certain parts of arith- metic. Equivalents will be found in terms of classes and relations between them for the G. C.D., L. C. M., the theory of divisibility, the theory of congruences and unique factori- ARITHMETICAL STRUCTURE. 165 zation into primes in the sense of rational arithmetic MN. There is also a theory of forms for classes but, owing to the law of tautology, it is so rudimentary as to be negligible. The quotient in ¥ is not unique, that is, the algebra from which the arithmetic is constructed is not a division algebra, although (rather remarkably) the fundamental theorem of arithmetic holds. At least two methods of isolating those parts of the algebra of classes which are abstractly identical with parts of arithmetic are available. We may first seek a theory of divisibility and then construct a theory of con- oruences consistent with it, or we may take these steps in the reverse order. A chief objection to the first is the absence of a set of postulates broad enough to include all known instances of integral elements. By the second method, provided we can construct a theory of congruences, we are ipso facto given at least one possible choice of integral elements. We are not however thereby given a theory of residuation in the form of the division transformation which is the foundation of Euclid’s algorithm for the G. C. D. The absence of the division transformation in a given theory however does not necessarily show that the theory is non-arithmetical in all respects; for example there may be a multiplicative theory, as is indeed the case for Dedekind ideals, where the trans- formation is lacking. According to the specific interpretations of the elements concerned one of the methods will usually have decided advantages over the other. When the elements are classes the methods are on a level, owing to the extreme simplicity of the algebra underlying the arithmetic. Hence in the present instance we may choose either; on account of its greater novelty of approach we shall first seek a theory (there is more than one) of congruence for classes. ARITHMETIC &v OF &, §§ 3-7 3. Arithmetical congruence. For clearness we state the postulates for a ring R that will be used. Consider a set 2 of elements x,y,z, -..,u’, 2’ and two operations S, P (— addition, multiplication) that may be performed upon 166 ALGEBRAIC ARITHMETIC. any two equal or distinct elements x, y of 2, in this order, to produce uniquely determined elements S(x, y), P(x, y), such that the postulates (3.1)-(3.3) are satisfied. Elements of = will be called elements of SR. (3.1) If x, y are elements of R, then S(x,y), P(x, y) are uniquely determined elements of R, and S (x, Y) - Sy, z), Ply, Y) = Ply, z). (3.2) If x, y, z are any three elements of R, then S(S (x, Y), 2) = Sz, S(y, 2)), PP, Y), 2) = Plz, Ply, 2)), Pz, Sy, 2) = SP, Y)s P(x, 2). (3.3) There exist in R two distinct unique elements, denoted by u', 2’, called respectively the unity, zero of R, such that, if x is any element of R, then Sir, 2) ==, Plrw) =n. Next, consider a uniform relation C(x, y) in R, that is, C(x, y) is uniquely significant for each pair (x, ) of elements x, y in R, such that the postulates (3.4)-(3.7) are satisfied, where z, y, z, w are any elements of R. (3.4) Cl, y) -D- Cly, 2), (3.5) Cl, y)- Cly, 2) 1D: Cla, 2), (3.6) Ole, y) - Chez, wy 1: CAS(z, 2), Sly, vw), (3.7) Cl, 3): Cle, w) 1D: C(Plz, 2), Ply, vw). The dot between relations or propositions signifies as usual the logical “and”. We shall call C' algebraic congruence in R. Let MN denote rational arithmetic. If a, b, m are integers, an instance in RN of C is C(a, b) = (a = b mod m), but C is not sufficient to define in Jt the usual meaning of congruence. In addition we require ARITHMETICAL STRUCTURE. 167 (3.8) (a = 0 mod m) :=: m|a, mFO, B39) (ka=kbmod m) D («a = b mod w'), m0, where m | a is read in 0 “m divides a”, and where gm’ = m and gq = the G.C.D. of I, m. In these the sign = of arithmetic congruence will not be confused with = meaning definitional identity and :=—: meaning formal equivalence. In N, (3.8), (3.9) may be replaced by any transforms of themselves by formal equivalence. We shall not distinguish such transforms in the determination of a doctrinal function. Any set of propositions (or postulates) abstractly identical with (3.8), (3.9) and an instance in NR of (3.4)—(3.7) will be called arithmetic congruence for the elements concerned. The elements of ZL (the algebra of logic) will be denoted by Greek letters; thus eo, 8, y, -.. denote classes The null class (= the zero of ¥) is », and the universal class (= the unity of ¥) is &; the (logical) sum, product of any two ele- ments «, 8 of ¥ are written as usual «+ 8, a 8 respec- tively. The supplement of any element of ¥ is indicated by an accent; thus « is the supplement of «, and « is the unique solution of ¢-t+e&¢ = ¢ ad = w. It is assumed that if «, 8 are amy elements of £, then so also are «, aB, a+ B. We proceed now to solve (3.4)—(3.7) in ¥. That is, we shall find in ¥ an instance of the doctrinal function defined by (3.4)-(3.7). By evident analogies with the theory of division for Dedekind ideals or for Kronecker modular systems we are led (among other possibilities) to the following. By «|B in & we shall mean that « contains 8, that is, each element of # is in «. Then an instance of (3.4)—(3.7) is given by either of (3.10) Cla, 8) = afm, (3.11) Cle, f) = p|(n+ 8), in which p is an arbitrary constant element of €. Accor- dingly, in analogy with 9t we may write (provisionally only, 168 ALGEBRAIC ARITHMETIC. since the equivalents in € of (3.8), (3.9) in MN are yet to be satisfied), 8.12) (e = 8 mod p) =: aB|y, (3.13) (e = 8 mod pu) :=: p|(a- 8). That two solutions of (3.4)—(3.7) must exist in £, if one does, is evident from the dualism between addition and multiplication in ¥. This has no analogue in 9%. Either of (3.12), (3.13) may be taken as the solution of the problem of algebraic congruence in ¥. To obtain arith- metic congruence in € we must satisfy also any pair of equivalents of (3.8), (3.9), and for this we require the following considerations. 4. The arithmetic zero, unity in &; arithmetic division, addition and multiplication in £. We ex- clude division by zero in RN except in the one case when the dividend is zero, when, we shall say, the quotient exists and is indeterminate. This of course is not equivalent to saying that the quotient does not exist. If the quotient by zero is defined never to exist (which again is a radically different assertion from that which states that the quotient exists and is wholly indeterminate), we are forced into irre- concileable contradictions between 2 and £, for ow in £, while, according to the usual (loose) convention regarding division by zero in %N, 0/0 is without meaning. Our con- vention so far as IN alone is concerned alters nothing that is customary in MN; with regard to ¥ it makes possible a complete isomorphism. This perhaps is a minor point, but for exactness it must be stated. Consider in R a relation D such that (4.1) xD, (4.2) 2 BDy-ylDz 12: 2Dz, 4.3) 2 Dy yu x = y, where x Dy is uniquely significant for each x 4 2’ (2/ = the zero In RN) and y in RN, with the exception that 2/ Dz’ is ARITHMETICAL STRUCTURE. 169 significant but indeterminate in R. An instance in RN of (4.1)-(4.3) is given by zDy = x divides y. This solution is valid also in ¥. I for z+ 2 the truth value + of x Dy implies the existence in R of a unique w sich that y = Plz, w), the quotient in R is said to be unique, and similarly in any instance of R. The solution in N is unique; in 2A it is not. This is in fact the distinction between a holoid and an orthoid realm. But, according to our provisional descriptions of arithmetic it is immaterial whether the quotient be unique provided only that the fundamental theorem of arithmetic subsists. In & division as defined in a moment does not yield a unique quotient, but it does lead to unique factorization in the sense of MN. Analogy with the theory of ideals suggests that we take in ¥ either of the following, (4.4) aDB = a|p, (4.5) aD —= Blea. Thus (4.4) states that in 8 « divides 8 is identical with « contains B; (4.5) asserts that « divides 8 is identical with AB contains «. As in determining CO («, 8) a twofold solution (if one exists) is necessary by the dualism in &, and either implies the other. It does not yet follow that either of (4.4), (4.5) is an interpretation of division in ¥ which is consistent with algebraic congruence in €. To complete the solution we require the G.C.D., the L.'C. M., and the zero, unity in 2. The G. C.D. and the L. C. M. are given by the following abstractions to 'R of the G. C.D. and L.C. M. in R after transformation by formal equivalence as suggested by the theory of ideals, (it is obvious that these functions can have no meaning in terms of order relations if they are to be significant for MR, ¥ as well as Jt; hence the transformation). 170 ALGEBRAIC ARITHMETIC. Consider in R two operations G, L upon elements of R such that, x, y being in R, G(x, y), L(x, y) are uniquely determined elements of R and the postulates (4.6)-(4.14) are satisfied. (4.6) Glew), Liz, 3) are unique, (4.7) G(z,y) = Gly, =), 48) GCG Gly, 2) = G(G(x, y),2 = Gy, 2), the last of which defines G(x, y, 2); (4.9) Gz, y) Dz Gx, y) Dy, (4.10) 2 Dx. 2Dy 1D: 2 DC x, y); (4.11) Lz, oy) = Ly, 7), : (4.12) Liz, Lily, 2)) = L(L(n, 9,0 = Lim, 3,2), (4.13) 2D L(x, -yDIliz, uy), (4.14) z2Dz-yD: 2D: La, 9) De. For example, (4.13) asserts that in R, x divides the L function of x, y and y divides the L function of z, y. It would be more consistent to write D (x, y) for = D y, but we have chosen the form used in order to recall its interpretation in N. Clearly the above are satisfied in Nt by @ (a, b) = the G. C.D. of the integers a, b, and L (a, b) = their L. C. M. In ¥ they are satisfied by either of the following, in which it is obviously necessary to take account of the twofold solution for D in £, (4.15) aD = «|B, Ge, 8) = a8, L(e, 8) = op, (4.16) a DA=2S|n, Gla Br= af, Lin BY=a+t8, either of which follows from the other by the dualism in 2. It is now apparent that the addition «-}8 and the multi- plication «8 of ¥ are sufficient but not necessary for an arithmetic «x of ¢. In &y we shall define the zero { and the unity v by (4.17) Se, 0) = a, Pla, v) == a, ARITHMETICAL STRUCTURE. 171 where S, P are as in either of the following, (4.19) Se, 8) = a8, Ple,f) = af, (4.20) Se, 8) an ap, P(e, 8) = a8, and therefore (5, v) — (mw, €) in (4.19), while (§, v) = (¢, ) in (4.20). The unity in 0 is the unique element which divides each element of W. In abstract identity with this we have in Lu, (4.21) aDp == «|B, y= &, vDy, (4.22) aD = Blu, ves wm, uD, where y is any element of ¥. Again, it is well known (Principia Mathematica, 1st edition, vol. I, p. 232, *24.13) that w|®w. In £5 we have, abstractly identical with R, {lr-GF0-C = wm), 5-H D-C = 9), are false propositions, 5. Recapitulation. We define nx by either of the following columns (which are duals of one another in &). Sum, S(e, 8): Cat 8; ap; Product, P(e, 8): ap; a+ 8; GC D, Glee, B): aitpB; ap; LCM, Lic 8: af; a+ 8; a = fBmod p: plletB); ably; zero, G: w; £: unity, v: £; ow; a divides 8, a DB: «|B; Ble, in which «, 8 are any elements of £; & ® are the unity, zero in &, and «|@ is read, “« contains, or includes, 8”. Either column gives an isomorph £y of 9 in ¥ when, as presently, we complete congruence. Order relations in Ot are replaced in Ly by the following. If in a given set of elements of x there be a unique element different from the unity in ¥y which divides each element of the set, it is called the lower extreme of the set; if in 172 ALGEBRAIC ARITHMETIC. a given set there exists a unique element different from the zero in ¥y which is divisible by each element of the set, that element is called the upper extreme of the set. Division is of course to be taken in the sense of a definite one of the above duals, and the zero, unity in Ly are taken from the same one. By the use of extremes “greatest” and “least” in the G. C.D. and L.C. M. can be restated in exact identity with 9%. For example, the “greatest” = the upper extreme, the “least” = the lower extreme, and the G. C.D. then becomes the “greatest” element of 2x which divides each element of a given set; the L. C. M. becomes the “least” element of ¥y which is a multiple of each element in a given set. It is to be noticed that “greatest”, “least” are not necessarily identical with “most inclusive”, “least inclusive” respectively; the roles with respect to inclusion may be the exact opposites of these. Further, the property in MN that the product of the G. C.D. and L. C. M. of two elements is equal to the product of the elements is preserved in Ly (either type as above). 6. Arithmetic congruence in Ly. It is now clear that (3.8) in MN has in ¢y the equivalent (6.1) (c = £ mod 2) i=: yD, To find the equivalent of (3.9) we need the concept of residuals as used in modular systems; the residual also exists in 0. We shall first define it for R. If a, b, 7, m are elements of MR, in which the unity is «’, such that m is uniquely - determined by (6.2) [aD{P(, b)}]-lmDI]-[m + u'], (that is, if « divides the product in R of / and b, and m also divides /, and m is different from the unity in R), where / runs through all elements of R, m is called the residual bRa of b with respect to a, and we write m = bh Ra. In 9 the residual of & with respect to m is the quotient of m by the G. C.D. of k and m; hence it is m’ in (3.9). In Ly, (6.2) becomes ARITHMETICAL STRUCTURE. 173 (6.3) [«D{PC, A] -[wDA]-[uFv]:i=:p — BRe, where / is an arbitrary element of £y. Hence, in 8, the abstractly identical equivalent of (3.9) in M2 is (6.4) |[P(x,0) = P(x, 8) mod y] D [a = 8 mod zRu], which can readily be verified to be a true proposition for either form of Ly. 7. The fundamental theorem of NM in Ly. As re- marked in Chapter I it is frequently profitable in seeking a unique factorization theorem to follow up any property of uniqueness for the elements considered. For example, from the unique expression of any symmetric function of given elements of 20 in terms of the elementary symmetric functions of those elements we reach at once an isomorph of the multiplicative part of 9t for symmetric functions, to which € and its consequences can be immediately applied. For x a sufficient property is given by Boole developments (Laws of Thought, Chapter V, especially Prop III; also Whitehead, Universal Algebra, Chapter II). All terms in a given deve- lopment having zero coefficients are assumed to have been deleted. The product in £ of any two terms in a given de- velopment is the zero in ¥; the sum of all the terms is the unity in £. Hence if a, 8 are any distinct or identical (in which case « = 8) terms in a development, « |8D x = 8. From a given set of classes is generated by the operations of logical addition, multiplication and taking of supplements; the Boole development of the (logical) unity of the set gives a set of terms such that the development of any element of ¢ as a sum in & of such terms is unique. The dual devel- opment is also unique and is obtained by taking supplements of both sides of the original development of the supplement of the given element of ¥. These considerations give us the fundamental theorem of N in Ly. It is to be understood that addition, multipli- cation in Ly refer to a definite one (either) of the columns in § 5. Tt is then easy to verify the following, where we 174 ALGEBRAIC ARITHMETIC. have arranged abstractly identical theorems and definitions from MN, Ly in parallel columns to show at a glance the isomorphism. % 1) the G. C.D. ofa, b is 1, then a, b are called coprime. (7.21) If k divides the product of « and b, and %k, a are coprime, then % divides b. (7.31) ¢ is called prime if k+ 1 divides ¢ when and only when k = g. (7.41) Primes exist; they may all be found by sifting (Eratosthenes), and they form a coprime set. (7.51) A positive integer is uniquely the product of primes. (7.61) The G.C.D. and L.C. M. of any set of positive integers can be written down from their resolutions asin (7.51). (7.71) By algebra € the mul- tiplicative properties of arithmetical functions are reduced to abstract identity with 2. Xn (7.12) If the G.C.D. of «, 8 is v, then «, 8 are called coprime. (7.22) If = divides the product of ¢« and B, and 2, « are coprime, then x divides A. (1.32) mw ‘is ‘called prime if and only if (x Dn). (x Fv) Yi (7.42) Primes exist; they may all be found from the Boole development of {, and they form a coprime set. (7.52) A given element of Ly is the product of prime elements in one way only. (7.62) The G.C.D. and L.C. M. of any set of elements of Ly can be written down from their resolution as in (7.52). (7.72) The same as (7.71), with obvious changes in notation. The list can be indefinitely extended and we have already seen that the theory of congruences in 9 goes over into Ly. In summary we can state that the theory of class inclusion and that of congruences and divisibility in rational arithmetic are distinct values of one doctrinal function. The dual solution in § 5 abolishes any intrinsic distinction between “least” and ‘greatest’ in the senses of .,least in- ARITHMETICAL STRUCTURE. 175 39 i$ clusive”, “most inclusive”. Accordingly, when in the following we refer to G. C. D’s and L. C. M’s it is to be understood that our universe of discourse is a particular one of the columns in § 5, and that, throughout a given context, the same one of those columns is meant. The possibility that in a given theory sometimes one interpretation is used and sometimes the other, according to the end in view, may be ignored, as it leads to nothing essentially new, as can be easily shown. ! ARITHMETIZATION, §§ 8-9 8. Classification of doctrinal functions. According to current definitions a doctrinal function is a set of postulates together with the set of all logical consequences of those postulates; a postulate is a propositional function, that is, the symbols of relations and elements (relata) are marks, and the relations and relata are variables. If specific inter- pretations can be assigned to the marks so that the resulting doctrinal function has the truth value -, we shall call the result (as before) a value or an instance of the function. In our present discussion we are interested in the functions themselves, not in their values. Let C(7) be a doctrinal function. By the above (usual) definition C'(7') contains in general parts which are not logically independent of other parts. If from C(7T) we isolate C, (7) such that C, (1) D C(T) is true and G(T) DC, (TY is false if C,(7) and CO, (7) ate distinct, where C, (7) is a part of C(7), we may call C, (7) a reduced form of C(T), and we shall assume that if several reduced forms of C(7') exist they are logically equivalent. This amounts to replacing C(7) by its set of postulates. Henceforth we assume all C(7)’s to be reduced; a C(T) is thus a class of propositional functions. Hence to the set of all C(T)'s we may apply &n, resolving each C(7) into its prime factors and thence determining the G. C.D and L. C. M of two or more. Thus ¥y gives us a means of classifying general theories with reference to their relations of inclusion with respect to implications, and this classification is abstractly 176 ALGEBRAIC ARITHMETIC. identical with the multiplicative properties of the positive rational integers. The algebra € therefore is applicable to the study of abstract structure. It is not necessary, of course, in any of this, that the 7 from which C(7) is constructed shall have any numerical significance. It is therefore perhaps not too much to say that the theory of logical structure is an instance of certain parts of N. 9. Nature of arithmetization. From a given theory © we construct its C'(@), and we assume that we have already constructed C(N) (N = rational arithmetic). Let the G. C.D in L of C(0), C(N)be G. From G we have C(G). Let the result of replacing in C'(¢) all marks (of relations and relata) by their instances as in C(®), be C(&'). Then if C(O’) is a value of C((), we shall say that @ is arithmetized lo the extent O' or that © is the arithmetic of ©. This provides for the case where & has no arithmetic. In the above we have taken N, rather than any of its current (partial) generalizations as the type of arithmetic. If it be desirable to replace WM be any of its partial gene- ralizations, the procedure with respect to these is the same as with respect to NX. It would be of interest first, however, to determine to what extent the existing extensions of Jt are themselves arithmetic in the sense of C'()N), for it seems that the ultimate difficulties of arithmetic reside in the natural numbers rather than in their extensions. Addition, 5. algebra, common, 5. arithmetic, 162-163. — additive, 12. — multiplicative, 12. arithmetical theory, 11. — complete, 11. — improper, 11. — proper, 11. restricted, 11. arithmetization, 176. Characteristic, 108. — even, 108. — odd, 108. class numbers, 101-106. composite, 113. — B. 117. congruence, 165. — algebraic, 166. — arithmetic, 167. conjoint, 31. — zero, 31. conjugates, 32. conjunction, 31. coprime, 30. Decomposition formulas, 48. — in PB, 50. degree, algebraic, 63, 72. = even, 40. odd, 40. — of functional product, 110. of parity function, 40. differentiation in €, 29. INDEX (Numbers refer to pages) 177 | Elements, 6, 8. — equivalent, 9, 10. elements, indecomposable, 9. — integral, 124. — irregular, 6, 18. — regular, 6, 18. — special, 18. Field, 5. — abstract, 5. — irregular, 5. — C-matrie, 19. — D-matrie, 19. — umbral, 155. form, 80. = even, Si. — odd, 51. — of order k, 80. function, 11, 16. arbitrary, 107. associated, X, C, D, 20. base of primary, 116. circular in A, 77. codivisor of, 57. composite, 123. comultiple of, 57. derived, 114. divisibility of, 56. equality of, 56. tactorable, 127. G. C.D. of, 58. L. C.M. of, 58. multiple of, 56. numerical, 144. parity, 66. 178 function, primary, 113, 115. — prime, 123. — quasi-even, 108. — quasi-odd, 108. — rational, 123. — reciprocal, 123. — type of, 69. — uniform, 16, 115. — vanishing over matrix, 55. General theory, 161. generator, 121. — equality of, 121. — fundamental, 123. — integral, 124. — rational, 123. — reducible, 123. Identity, abstract, 5. index, 158. instance, 5. integer, simple 127. integration in €, 29. invariants, 92. isomorph, 44. Manifoldness, 128. matrix, 15. — absolute product of, 16. — algebraic, 122. — (0, 15. — conjoint, 31. — coprime, 30, 41. — D, 15. — equality of, 15, 16. — in BB, 15.. — normal, 15. — order of, 15, 16. .— partition of, 33. — scalar product of, 16. — transcendental, 122. = zero, 10. module, 7. multiplication, 5, 8. — (¢, 19, 20. INDEX. multiplication D, 19. — of parities, 42. — multiplicity, 128. Negative, 6, 19. Order, compound, 83. — even, 40. — of matric variable, 17. — odd, 40. — of parity function, 40. Paraphrase, principle, 67. — extended form of, 79. — extension to higher forms, 80-88. — integration of, 87. — modified principle, 68. parameter, 20. parity, 36. — absolute, 37-39. — even absolute, 36. — multiplication of, 42. — odd absolute, 36. — product of, 42. — relative, 37. — relative coprime, 41. — transform, 62. parity function, 66. — restricted, 66. partition, 33. primary, 113. — form, 113. product, 5, 6, 8. — absolute, 16. — «7 19, 90. — DD. 19. — FK, 118. — matric, 31. — partial, 19. — of parities, 42. — sealar, 16. — of sets, 56. — symbolic, of functions, 109. Quasi-constant, 109. quotient, 7. Ray, 7. reciprocal, 7, 8, 19. relation, uniform, 166. representation, 81. — compound, 81. — compound limited, 87. — limited, 85, 86. — limited compound, 87. — as sums of squares, 103. residue, positive, 32. ring, 7. — modified, 114. Scalar, 16. — instance, 149. — product, 16. semigroup, 8. — arithmetical, 11. — associated, 23. — commutative, 8. — improper, 11. — proper, 11. series, exponential, 29. — power, 28. sets, 15. — codivisor of, 56. — comultiple of, 56. — division of, 56. — equality of, 56. — G.0.D. of, 57. — L.C.M. of, 57. — product of, 56. — sum of, 56. — total, 56. simple, 127. substitution groups, 92. sum, 5, 6, 32. —(, 19, 20. — D, 19. — F, 118. — matric, 31. INDEX. 179 sum, partial, 19. — of sets, 56. Theta, 64. — quotient, 64. — of p arguments, 106. trace, 32. transform, 62. transformation, 62. — identical, 49. — parity, 62. Umbra, 29, 147. unit, 9, 10. unitary, 83, 86. unity, 5, 8, 18, 118. Variable, 17. — coprime matric, 30. — integral value of matric, 80. -— matric, 17. — scalar, 17. — value of, 17. variety, 5. — functional, 21. — taken over A, 26. — WU; 5. — B14 28. — 9B. 3, 27. — B, Un, 26. — By.p Un, 26. — €, 13, 21. - G,.Y, 21. — Cp Un, 19, 23. — Ex, Un, 19. — BD, 13, 28. — Dp Un, 24. — Dp Un, 19. — (E185 112. 919. — E,, 125. — &, Fer Er, D. — 6, 8, 121. — Be, 28. — Sp, 22, 28. — Og, 128. 180 variety Ox, 22. 33. 6. SU, 155. g, 164. 2x, 184, 170. Mm, 7. — M, 160. — 5, 14 34 46. R. 7, 160. = Ro, 44 Se, 24. — Gp, 2. — 3, 44. INDEX. | variety %., 150. = 1.5. — Un, 5. — WN... 6 — 8B, 5. — XW, 23. | — %,.0Us, 23. | — %,1,, 20. - 1%... 0, 2 — 9. U0, 27. | Ea Vo 0X; 27. | Zero, 5, 18. mI 7 DAY USE RETURN TO DESK FROM WHICH BORROWED ASTRON., MATH. STAT. LIBRARY : General Library RB 17-60m-8, 60 (B3395s10)4188 Uaiversiy of California wii cn374a8909d Ft SR | P : 2% Bhim