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VoL 1, No, 1, pp. 1-44
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AN ARITHMETICAL THEORY OF CERTAIN
NUMERICAL FUNCTIONS
by
ERIC TEMPLE BELL
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Vol. 1, No. 1, pp. 1-44 August, 1915
AN ARITHMETICAL THEORY OF CERTAIN
NUMERICAL FUNCTIONS
by
ERIC TEMPLE BELL
SEATTLE, WASH.
PUBLISHED BY THE UNIVERSITY
1915
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AN ARITHMETICAL THEORY OF CERTAIN NUMERICAL
FUNCTIONS.
BY E. T. BELL.
0. PRELIMINARY CONSIDERATIONS.
0.00. Apart from any evident utility as an economizer of thought and of
calculation, there is, in the manifold interpretation of a system of postulates
a wide philosophical significance. 1 The numerous instances of this multiplicity
of meaning that have been devised in geometry, are common knowledge; in
arithmetic the comparatively fewer examples, among which the Theory of
Ideals of DEDEK,IND is the classic, do not seem to be so generally appreciated,
possibly because they lie slightly to one side of the main progress of analysis,
although, as asserted by some, 2 arithmetic may be the proper foundation of
all. The purpose of this paper is twofold; (i) To show, by several examples,
that the postulates and processes of arithmetic admit of a multiplicity of
interpretation, all examples to be simple and interconnected; (ii) To construct
a self-contained arithmetical theory [cf. 0.01 (i)] of a large and important class
of numerical functions, the theory to be so formed that the inter-relations of
the functions considered [those in 5.23], shall be exhibited with a minimum of
calculation, from either their symbolic or verbal definitions. Up to a certain
point (i), (ii) are identical; beyond this, certain class-properties of the func
tions must be imagined in order to complete (i), and these newer aspects of the
functions are relative only very remotely to the integers. The object in
this part is to carry the developments sufficiently far to support several
distinct interpretations of arithmetic [the simplest of all in this paper is in
4], and the studies of congruences and forms [11] for lack of space, have
been deferred, although the methods in which they are to be approached are
indicated. In all, sufficient material is provided for 32 distinct duals of
arithmetic; in a narrower sense, it is shown [6.35], that an infinity of such
may be evolved from a single mould. In the illustrations of (ii), given in
7, 8, only those which may be briefly written, and not the most interesting,
have been selected from a great quantity found by the methods of the paper.
The more interesting applications arise from the consideration of (ideal)
forms [ 10]. Throughout, the insistence is upon methods rather than upon
details of calculation which are elementary in all cases, and may be easily
supplied if not indeed obvious.
1 Cf. CASSIUS J. KEYSER: Concerning Multiple Interpretations of Postulate Systems and
the "Existence" of Hyperspace. [Journ. Phil., Psychology and Sci. Meth. (New York), Vol.
IX, No. 10, 1913.]
2 E. g., by KRONECKER.
532156
2 AN ARITHMETICAL THEORY
The properties of an integer may be regarded in two essentially distinct
ways; either in relation to itself and its separate divisors, or in relation to all
integers; these may be called the static and the dynamic aspect respectively,
relying on an obvious analogy. Of these, the static is the more easily
investigated aspect; but, if (i) is to be accomplished, then clearly the dynamic
must also be considered. In relation to (ii) the dynamic properties of integers
are not taken account of, principally because of inherent difficulties that seem
to place such considerations beyond the present reach of analysis, but also,
because the theory of numerical functions in their dynamic aspects ceases
to be arithmetical in the sense of [0.01], or in that well-expressed by E. CAHEN:
"In algebra, division is only exceptionally impossible, in arithmetic, only
exceptionally possible."
Throughout, *, t refer respectively to definitions and theorems; thus *
indicates that a section contains a definition essential to (i), f, that the section
contains a theorem.
*0.01. (i) For precision in regard to 0.00 (i), an arithmetical theory may
now be defined. It is emphasized that throughout, addition, subtraction,
multiplication and division, are used in their abstract meanings, that is, as
operations which obey the " ordinary abstract operational laws of algebra." 1
With respect to these operations, 2 a number system, N , that is also a field 1 is
postulated. Let now a system of elements N be defined similarly to N in
all respects, except that in N there are n identity elements 2 with respect to multi
plication. These n elements are the units in N. Elements of N that differ
from one another only by unit factors are equivalent, and are considered not
distinct. 3 Then, N is defined to be an arithmetical field, when and only when
(1) and (2) hold:
(1) The assemblage of all elements in N is denumerable; and hence, if n
is infinite, the units form a denumerable assemblage.
(2) Within N there is a denumerable assemblage, P, of elements, which are
such that with respect to elements of P every element of N satisfies the funda
mental theorem of arithmetic, viz., every element of N may be represented as a
product of powers of distinct members of P in one way only. [Extensions
will be dealt with as they arise naturally; e. g., in 4.64.]
(ii) The totality of relations between all members of an arithmetical
field, the relations being with respect to the (abstract) fundamental laws
of algebra, constitute an arithmetical theory.
(iii) Properties of the elements of N that are immediate consequences of
1 E. H. MOORE: Bull. Am. Math. Soc., vol. Ill (1893), p. 75. The order of the field is
here infinite; if finite, a finite arithmetic is similarly, mutatis mutandis, defined; but finite
arithmetics are not considered in this paper.
2 For a very clear statement of what constitutes a number system, cf. O. VEBLEN and
J. W. YOUNG: Protective Geometry, vol. I (1910), p. 149. Cf. also J. KONIG: Algebraischen
Groszen (Leipzig, 1903), pp. 1-9.
3 For the precise statement of the senses in which the units of this theory are respectively-
identical or distinct, cf. 3.29; 3.30; 3.31 and 6.03; 6.07; 6.09; 6.10.
OF CERTAIN NUMERICAL FUNCTIONS. 3
the fundamental theorem of arithmetic, or which ultimately depend upon
that theorem, are arithmetical.
(iv) If to (1), (2) be added further postulates, the whole being consistent,
there results an extended arithmetical theory.
0.02. As arithmetic proper is only concerned with the properties of
positive integers, so here, the main interest will be in the analogues of the
integers. The number of units in this theory is infinite, but they preserve an
important characteristic of unity, viz., their numerical values are identical.
There is no concept of magnitude in the elements of this theory; hence, as
the theorem that an integer is divisible by only a finite number of distinct
integers, must here find an analogue, the finiteness of an element is defined
otherwise than by its magnitude [cf. 6.13], but in a way equally applicable
to the integers. Neither is there any concept of a natural order for the ele
ments; an artificial order, having the properties in respect to the elements
that the order of 1, 2, 3, , has to the integers, may be defined in connec
tion with addition. Similarly for other concepts of arithmetic; these remarks
indicate in what respects the theory of certain numerical functions will be
shown to be isomorphic to arithmetic.
As addition, etc., are used often in the same context with abstract addition,
etc., where necessary to distinguish between the kinds, that proper to this
theory will be called ideal addition, etc. Moreover, in order to emphasize
the abstract identity of certain of the operations used, with +, etc., the sign
+ is sometimes used for a concept that arithmetically has the properties of
X [cf. 9]; this however is primarily done in accordance with usage. 1
*0.03. A function, f(x), is numerical, if f(x) exists for every positive
integral value of the argument, x: and moreover is such that /(O) = 0; /(I) = 1.
By convention, a constant is a numerical function.
*0.04. A numerical function f(x) [0.03] is factorable, if for every pair,
HI, n 2 , of relatively prime values of the argument, /(ni)/(n 2 ) = f(nin 2 ).
*0.05. The arithmetical definition of a numerical function, f(x), is the
statement of the values assumed by the function according to the classes of
values of the argument for which f(x) exists. [For examples, cf. 2.01; 2.02;
2.04; 3.14.]
*0.06. If f(x) is a numerical function, / is a functional form: and / is
abstracted from f(x) .
0.07. The elements upon which 0.00 (i) is constructed are the functional
forms of certain factorable numerical functions. The meaning of a functional
form, which is, in effect, the arithmetical definition [0.05] of a numerical
function, will be readily understood after the definition of ideal multiplication
[3.07].
0.08. Sections 1, 2 are in a sense preliminary to the main part, which
1 In the class-properties that are arithmetical, complete conformity with the rest of the
theory may be attained on representing logical addition by X ; but as this is in violation of all
precedents, it has not been done.
4 AN ARITHMETICAL THEORY
continues with 0.00 (i) in Section 3 " Ideal Multiplication." 2 is primarily
a source of future illustrations; and the purpose of 1 is explained in 1.00.
1. SIMPLE NUMBERS.
1.00. There is a gain in conciseness in many subsequent theorems if an
integer is regarded as a product of simple [1.01], instead of prime, numbers.
Also, it will be shown that any theorem which depends ultimately upon the
unique factorization theorem of arithmetic, is, according to respective resolu
tions into prime and simple numbers, susceptible of a dual interpretation
[6.34; 6.35].
*1.01. A positive integer that is divisible by the square of no prime, is
simple. Relatively prime simple numbers are distinct. Unity is simple, and
other simple numbers are denoted by Pi (i = 1, , oo).
1.011. If n is resolved into its prime factors, this is also a resolution into
simple factors; excluding this (unless n be simple, in which case the following
resolution and the prime resolution coincide) :
.fl.02. Any number may be resolved into distinct simple factors in one
way only. For, let n = pi ai p 2 a *- - -p r ar be the resolution of n into prime
factors, the primes being so ordered that on ^ <*/ when i > j; and for s =i r,
let a;, (i = 1, - , s) denote all the unequal a t -, (i = 1, , r) ; also let a t - = ai,
a s = a r , and ai < a 2 < < a s - Then, if P t - is the product of all those
primes pi which are such that n/pi ai is, but n/pi ai+l is not, an integer, the
required resolution of n into its simple factors is obviously, n = Pi ai P 2 a2t -P/*-
The notations of 1.02, 1.03 prevail throughout the rest of this section, and the
Pi, which are all distinct, are the simple factors of n. If n = TT^^TT^- "ir,?*
represents either the resolution of n into its prime, or into its simple, factors,
in the former case the TT S are all distinct and relatively prime, in the latter,
the jS s are all distinct and the TT S relatively prime, hence distinct [1.01].
*1.03. If ^ a/ ^ a t (i = 1, , s), and n = P/ TY 2 - - -P/ , then
n is a relative divisor of n. If 1 is the only relative divisor of n, n f , then n, n f are
relatively distinct; in symbols, Dv(n, n f ) = 1. If Dv(n\, n 2 ) = 1, then the
greatest common relative divisor of rm/, nn , is n; in symbols, if n\ = nn/,
n* = nnz , Dv(n\, n%) = n. Similarly the lowest common relative multiple of n\,
7i 2 , is defined by Lm(n\j n 2 ) = nn/n/.
The number, <p (n), of integers > n that are divisible by no simple factor
except 1, of n, is the relative totient of n. The number, and the sum, of the
relative divisors of n are denoted respectively by v (n) and <r (ri). In order
to indicate that n is being considered as a product of simple rather than of
prime factors, n is replaced by n ] numerically, n = ri. The function /* (w),
or n (n ) has the value if n is divisible by the square of any simple number,
and otherwise, is + 1 or 1 according as n is the product of an even or of
an odd number of simple numbers.
fl.04. The following are immediate consequences of the definitions;
(i) ( ) - K (a, + 1); (ii) , ( ) = H (1 -
OF CERTAIN NUMERICAL FUNCTIONS.
(iii) Dv(n, , n, )-Lm(n^ w 2 ) = mW; (iv) <p (n ) = n r
The last may be deduced similarly to the ordinary totient, <p(ri)j from the
principle of cross-classification of classes, 1 noting that in their totality the Pi-
are relatively prime [1.02].
fl.05. If Dv(ni t n 2 f ) = 1, and if $ (n ) is any one of the functions v (n r ),
<7 (O, v f (ri), their clearly, i^ OiW) = i> (nW(n* ). The like is not, in
general, true if Dv(n\ t n 2 ) > 1.
|1.06. As in the corresponding theorem for <p(ri), it may be shown that
Sv? (ff ) = n f , the summation referring to all relative divisors d of n .
fl.07. If $ (ri) is any function such that ^ OiW) = ^ / (n 1 / )^ (n 2 / ) when
Dv(n\ j n%) = 1, and if the summations refer to all relative divisors d of n f ,
then if * ri = S^ d is
The proof is similar to that usually given when the resolution of n (or n ) is
by prime, instead of by simple, factors. [Or, cf. 8.]
fl.08. All of the theorems in 1.04 to 1.07 are special cases of a general
result that now may be readily inferred; however, for the sake of uniformity
of treatment, a set of ideal elements, or l-numbers is introduced. Any par
ticular result of the above kind may be proved at once from first principles,
but the systematic finding of such is better done otherwise.
*1.09. The l-numbers 1, Z,- (i = 1, 2, -, oo) are such that Z 4= lj when
i 4= j, and are placed in (1, 1) correspondence with the natural primes, pi, so
that lij pi are correspondents. Hence, to any integer n = pi ai pz a -, ,
corresponds a definite symbol, /i ttl / 2 a2 . Recasting now the definitions in
1.01, 1.02, replacing therein any prime p t by l i} and defining any l^lf- -
as an l-symbol, the terms simple l-symbol and resolution of an l-symbol into its
simple factors, may be taken as defined; also, relatively distinct l-symbols are
now defined. The /-symbol corresponding to n is denoted by L(n f ), and
{L(n )} T = L r (n ). Thus a new set of L-numbers is defined, whose char
acteristic properties, and the only ones that are relevant for the present
purpose, are by definition; the product of any number of L-numbers is 1 or
according as the L-numbers are or are not relatively distinct in their totality.
Multiplication of L-numbers is commutative and associative: also, if m is any
integer, m X L(n ) = L(n ) X m.
1 1.10. Considering now the formal expansion of each factor, and sub
sequent distribution of the product, it follows from the definitions in 1.09
that S(s) = fl (1 ~ L(Pi)/Pi s )- 1 is equivalent formally to ] 1/n". For,
i=l n =l
any term in the distributed product is of the form L ai (Pi)L a > (Pj)- - -L ak (Pk)/
. 1 Cf. H. J. S. SMITH: On the History of the Researches of Mathematicians on the Subject of the
Series of Prime Numbers. Proc. Ashmolean Soc., Ill, 128-131; or Coll. Papers, I, p. 36. Or,
more briefly as in 8.
6 AN ARITHMETICAL THEORY
(Pi ai Pj a J -Pfc a *) s ; whose numerator is 1 when and only when the denominator
is the resolution into simple factors of some number, and otherwise is 0; also,
every possible product of powers of simple numbers, and hence every number,
occurs in the denominators. Hence if s is so chosen that the series converges,
the infinite product and series may be equated.
1.11. Most of the simple-number analogues of elementary arithmetical
theorems are readily suggested and easily proved; thus if n\, n 2 are relatively
simple, and if HI, n z f is relatively divisible by n , then so also is only one of
HI, n% . But sometimes the analogy is not so close; thus, if P is simple, PI
and P are relatively distinct. It suffices to show that if P = p\p>i- -p r , the
exponents of the highest powers of pi, p that divide P \ are unequal. Let
Pi < Pz, and let [m/n] denote the integral part of m/n: the exponents in
question are P/ Pi + [P/p*] + [P/p*] + ; (i = 1, 2). Since p, < p 2 ,
P/pi > P/pz , whence, for a ^ 2, [P/pi a ] = [Pjpf] , hence the exponents are
unequal. Usually it is sufficient to replace number, by simple number, prime,
by simple, but not always; e. g., <p (n r ) is not the number of integers > n and
relatively distinct to n. The exact meaning of relatively prime to n must be
transformed into not divisible by any simple factor of n . With such changes
the true analogue of any theorem is found without much difficulty; numerous
examples will be found later [cf. 6.34].
2. THE FUNCTION ^(n; a, b, c, I).
2.00. The majority of factorable numerical functions in current use, and
most of those used later for purposes of illustration, are specializations of a
single simple function, ^, [2.01]. In this SF, I = + 1 or 1, a, b are positive
integral constants, n is a variable integer, and c is finite, otherwise arbitrary.
This ^ includes the numerous functions considered by LIOUVILLE [ 7], and,
in its general form, wherein I is arbitrary, oo 4 in all. As defined in 2.01, ^ is
the simplest case of a more general function [7.02 (ii)], whose verbal definition
is so prolix, that it is best deferred until several new concepts shall have been
introduced [3, 5], All of these in turn constitute the simplest possible
example of a class of numerical functions, 1 which is the first in an infinite
assemblage of such classes.
*2.01. Two auxiliary functions are first defined:
(i) X(n); the multiplicity of n; that is, the total number of primes which
divide n.
(ii) y(ri); the manifoldness of n] that is, the number of distinct primes
which divide n. [This term is due to SYLVESTER.] Then V(n; a, b, c, I), for
I = 1 is defined by (iii) and (iv) :
(iii) ty(n , a, b, c, 1) = if n is not the perfect 6th power of a simple
number [1.01]; and in the contrary case, = c y(n) n a b .
(iv) ^(n; a, b, c, 1) = if n is not the perfect 6th power of an integer;
and in the contrary case, = ( c) x(nllb} n a!b .
1 Considered in 3.14; 5.24 (iii). The y(n) in 2.01 (ii) will not be confused with y(H) of 5.
OF CERTAIN NUMERICAL FUNCTIONS. 7
In both (iii), (iv), n alb signifies the arithmetical 6th root of n a . Clearly, if
n is simple, y(n) = \(ri).
*2.02. As a first specialization of ^, the functions on the left of the respec
tive identities are defined on referring to 2.01 [cf. also 2.05]:
t (n , a, 6) = (n; a, 6, - 1, 1)| x (n , a, 6) = * (n; a, 6, 1, 1)
l^ (n; a, 6) = *(w; a, 6, - 1, - 1)J x (n; a, 6) = * (w; a, 6, 1, - 1)
r (n; a, 6) = *(w; 0, b, a, 1)
r r (n; a, 6) = *(n; 0, 6, a, - 1)
Each pair is evidently a particularization of the pair ^(w; a, 6, c, 1), whose
fundamental property may be stated, although not proved until 7.04:
f2.03. The summation extending to all divisors D, of A", or to all relative
divisors D, of N, according as N is regarded as a product of prime or of simple
numbers, the value of
S9(D; a, b, c, Q9(N/D; a, b, c, - I)
is 1 or according as N = 1 or N > 1.
Putting I = 1, the meaning of the theorem is seen for the functions in 2.02.
*f2.04. Of the six functions in 2.02, sixteen sub-cases are of such frequent
occurrence in arithmetic, that they have received special notations, now given;
some of the symbols are due to LiouviLLE 1 and other writers on the subject,
but no attempt at conformity with these in all has been made, as the subse
quent point of view is distinct. Comparing with 2.02, 2.041 for verifications
of the implied theorems, the sixteen are [cf . also 7] :
(i) t(n; 1,2) = n( Vn)fc 2 (w)t*i( Vn).
(ii) \l/(n; r, 1) = n(n)ur(n).
(iii) V(n; 1, 1) = Ul (n).
(iv) t (n-, 1,2) *s
(v) f(n , r, 1) = ur(n).
(vi) x (n; 1, 1) = { M
(vii) x (w; 1, 1) = m
(viii) x (w; r, 1) = -ur(ri)u r (ri).
(ix) r(n; r - 1, 1) ^ iT-i(n) = {/*(n)}(r - l)" ( ->.
(x) f(w; 2r - 1, 1) - f 2 r-i(n) = {M(n)} 2 (2r - !)*<>.
(xi) f(n; 1, 1) = { M (n)} 2 .
(xii) r(n; 2, 1) s ( M (n)}Mw).
(xiii) r(n; - 1, 1) = M(W).
(xiv) f(n; 2^ - 1, 1) * |/*W} 2 (2 - 1)"^.
(xv) f (n; 1, 1) = -nr(n).
(xvi) r r (^; - i, i) = MO(W).
*2.041. The definitions of the various symbols on the right of the sixteen
1 References in 7.
8 AN ARITHMETICAL THEORY
identities in 2.04, are, for n a positive integer, r a positive integer unless the
contrary is expressly stated :
(i) ^(n) is the function of MOBIUS (sometimes, of MERTENS), and vanishes
if n is not simple, and otherwise is + 1 or 1 according as the manifoldness
of n is even or odd. Mi/aW = M( Vw), which exists only when n is a perfect
ath power; A/n = + A/ft, arithmetical root.
(ii) k r (ri) = 1 or according as n is or is not a perfect rth power.
(iii) u r (ri) = n r ; Mi(Vft) may be written uuz(ri), or u l P(ri)\ similarly for
higher roots.
(iv) nr(ri) = + 1 or 1 according as the multiplicity of n is even or odd;
viz., TD-(n) = (- 1) A <>.
(v) v(n) = the number of divisors of n. In addition to these, for com
pleteness;
(vi) <p(ri) = the totient of n: viz., the number of integers > , and prime to, n.
(vii) <r(ri) = the sum of the divisors of n.
(viii) 0(n) = the total number of decompositions of n into a pair of rela
tively prime factors; in this, if n = nin z is one such decomposition, ntfi\ is
to be counted as another, distinct from the former.
(ix) (D r (ri)} r = the greatest perfect rth power that divides n; written
D r ^(n). Among the divisors of n are included always, 1 and n; also, if any
of these functions is undefined (by its nature), or is ambiguous, for n = 1, by
convention, the value is taken to be unity.
f 2.05. Comparing the definitions of the functions in 2.02 with that of
the ^-function in 2.01, it is easy to verify that:
(i) $ (n; a, 6) = k b (ri) Hub(ri)-Uaib(n) ,
(ii) \l/ (n-, a, b) = k b (n) U al b(n).
(iii) x (n: a, b) = k b (n)- jjui/bW } 2 U a/b (n);
(iv) x (w; fl, 6) - (~ l) A < 1/6 >-Ww)-w,6(n).
(v) r (n; a,b) = WW} 2 -/Wn).a^>;
(vi) f (n; a, b) = (- 1)*( 1/6 ) -k b (n) .a A < 1/6 >.
The verbal equivalents are:
(i) \J/(n; a, b) = if n is not the 6th power of a simple number, and, in
the contrary case, = db n alb according as the multiplicity (or, what is here
equivalent, the manifoldness), of n l i b is even or odd.
(ii) ^ (n] a, b) = or n a l b according as n is not or is a perfect 6th power.
(iii) \(n\ a, b) = or n alb according as n is not or is a perfect 6th power
of a simple number.
(iv) x (^; a, b) = if n is not a perfect 6th power, and in the contrary
case, = n alb according as the multiplicity of n llb is even or odd.
(v) f (n; a, 6) = or a Y(n) according as n is not or is a perfect 6th power of
a simple number.
(vi) f r (n; a, 6) = if n is not a perfect 6th power, and in the contrary
case is d= a A(nl/6) according as the multiplicity of n l/b is even or odd.
OF CERTAIN NUMERICAL FUNCTIONS. 9
2.06. In the set of sixteen [2.04], (ix), (x), (xiv) appear but slightly
different from each other; when their inter-relations with the others come to
be examined, it will be found that they are essentially distinct. Anticipating,
if either a = b = 1, or if otherwise, a, 6 are unequal, it will be shown that the
six functions of 2.02, and hence the sixteen of 2.04, when considered as func
tions are ideal primes, although several, e. g., (i), (ii), (iv) of 2.04 apparently
contradict any notion of primeness. Also, v(ri), <r(n). <p(n), 6(n), and many
others will be exhibited as ideal products of powers of the sixteen primes. It
is the prime property that makes these specializations of ^ important for
illustrations, etc.
3. DEFINITION OF IDEAL MULTIPLICATION; ETC.
3.00. The concept of ideal multiplication may be briefly illustrated by
the first (historically) example in which it is implicit. Denoting by <p(n)
the totient of n (viz., the number of integers not greater than, and prime to, n),
and by u (ri) a numerical function of n which has the value unity for all values
of the argument n, and by Ui(n), a numerical function which has the value n
for all values of the argument n, there is the well-known theorem ^<p(d) = n;
or what is equivalent,
(i)
the notation , etc.; meaning (as customary) that the summation is extended
over all divisors d, including 1 and n, of n.
Modifying the notation, the equality (or identity) (i) may be rewritten,
<PUQ^UI , and is to be read, "the functional form u\ is equivalent, ~, to the
symbolic, or ideal product of the functional forms <p and UQ"; or, <p and UQ
constitute a pair of ideal divisors of Ui"; or, "ui is divisible by <p and UQ com
pletely"; and <pu ~ u\ is an equivalence of functional forms. Throughout, n is
any positive integer.
*3.01. If \j/, \l/ j \j/" are functional forms [0.06], which are such that for all
positive integral values of the argument n,
(i) T,t Wt"(-} = i(n)
co \ d )
the 2 notation being that of 1.00; the equality (i) being rewritten in the form
^Y" ~~ if, t each of the functional forms ^ , $" is called an ideal divisor of the
functional form \f/ } and \l/ is the ideal product (with respect to ideal multiplica
tion) of \j/ j \J/"; also, ^ is completely divisible by \j/ r and $".
Henceforth, if /i, / 2 are any functional forms, /i/ 2 shall signify an ideal
product as defined; viz., /,/ 2 shall mean ]C/i(d)/2 ( ^ ), wherein the argument n
(n) \ a /
is general. The sign "~" is read, "is equivalent to," and expresses the pre
cise relation above defined; ty ^" ~ ^ is an equivalence.
10 AN ARITHMETICAL THEORY
f3.02. As a relation, equivalence is symmetrical and transitive; viz.,
(i) if M" ~ ^, then ^ - i//i// ; (ii) if t ^ <A and ^ ~ ^", then ^ ~~ ,// .
(iii) ^V" ~ ^ V; that is, ideal multiplication is commutative. These are
obvious consequences of 3.01. The concepts of 3.01 are now made general 1
so as to apply to any number of functional forms, ^, \f/i, fa,
*3.03. Denote by di, d 2 a pair of conjugate divisors of n, so that n = d/d 2 ;
resolving similarly, let d z = d 2 d 3 ; - - etc.; whence, finally,
n =
d r <2 = dr2 d r l,
Let ^i(ni), ^2(^2), *! ^r(^r) denote numerical functions [0.03], and consider
the r-fold summation, S r (n), where
(ii) S r (n) ^
(n)
the summations on the right being over all values of the product $i(di)faW)
- $r(d r ) which are formed by supplying the arguments d/ to ^i(ni) in
^1(^1)^2(^2)- tr(n r ); (i = 1, , r) from all possible solutions of the set (i).
Then;
f3.04. S r (n) is a numerical function.
f3.05. If \l/i(ri) (i = 1, ,/) are factorable numerical functions [0.04],
then S r (ri) is a factorable numerical function.
3.06. The proofs of 3.04, 3.05, are immediately evident from the defini
tions.
*3.07. Considering 1 S r (ri), or for uniformity ifr(ri)j [ = S r (n)] as a numer
ical function of the general argument n, the identity in (ii) will be written in
the symbolic (or ideal) form,
(i) ^ ~~ ^1^2- &
Conversely, if a functional form ^ be given, and it is possible to determine
1 Throughout the entire paper, it will be well to remember a saying of EDOUARD LUCAS
[Theorie des Nombres, Paris (1891), p. 205]: "The symbolic calculus must be considered as a
rapid method for the writing of formulae in a series of theoretical deductions; but, whenever
it is a question of determining the values of the numbers furnished by this calculus, it is indis
pensable to replace the symbolic formula by the ordinary development. ... It is then, in a
certain measure, a shorthand of the formulae of arithmetic and algebra for the development of
new theories." The symbolic calculus developed here is, of course, distinct from that of
LUCAS, but the latter may be applied to the former, since, as will be shown, the theory of
numerical functions is isomorphic to both algebra and arithmetic, thus giving rise to new
functions or new properties of the present functions; but this does not properly belong to this
part. The remark of LUCAS is especially pertinent to 0.00 (ii).
OF CERTAIN NUMERICAL FUNCTIONS. 11
functional forms fa, ^ 2 , fa in such a way that 3.03 (ii) is an identity for
all values of n, then this fact is expressed by 3.07 (i), which is to be read:
the functional form ty is equivalent to the ideal product of the functional forms
ti> $2) ) i/v; or, the functional form ty is divisible by the functional forms
V lj ^2, , fa, completely; also, each of \l/i (i = 1, , r) is an ideal divisor of
\l/, and the operation of forming the product (ideal) of fa, fa, , fa is ideal
multiplication.
f3.08. Ideal multiplication is associative. It will be sufficient to show
that (fa fa) ^3 ~ fa(tzfa), the left of this equivalence denoting the ideal product
of \f/ s and the ideal product of fa, fa, the right, denoting the ideal product of
\f/i and the ideal product of fa, ^ 3 ; (for, by 3.02 (iii) ideal multiplication is
commutative). The theorem follows at once from 3.03 (i), (ii), remarking
that the totalities of all solutions of n = did 2 , di = di di", and n = 5i5 2 ,
5 2 = 8 2 f S 2 ", for any particular value of n, are identical (except for order), the
totalities being respectively all values of d\ t di", d- 2 and 5i, 5 2 , 5 2 " that satisfy
the equations, n = di d\"dz , n = Si<5 2 <5 2 .
3.09. Within the entire class of numerical functions, it is not difficult to
show [cf. 5] that, in the sense of 3.07, any functional form ty is divisible by
any other functional form \l/ f } moreover, it may be shown that if \J/, \J/ are given,
ty" may be determined in an infinity of ways so that ^ ~ ^ ^" , $, ^ , V being
numerical functions. Hence if an arithmetical theory [0.01] of the numerical
functions is to be constructed, their definition must be modified. It is the
purpose of the sections following to select from the entire class of functions
denned in 0.03 a denumerably infinite sub-class for which an arithmetical
theory exists. Henceforth, the juxtaposition of functional forms shall signify
the ideal product of the forms, and the qualification ideal may be dropped in
referring to multiplication.
*3.10. If \f/(n) is a numerical function which vanishes for all values of n
greater than unity, then \j/(n) is a unit function, and \l/ is a unit functional form;
briefly, ^ is a unit. [For examples, cf. 2.03; 7.04; 7.09.] Units will be denoted
by e, e , , ei, 6 2 , , etc.
*3.11. If \}/ is the product of several functional forms, of which at least
two are distinct from units, ^ is composite. [Example 3.00; u\ is composite;
but, cf. 3.35.]
*3.12. If for all positive integral values of n, 1 \[/(ri) = \l/ (ri)\l/"(ri) (algebraic
product), then ^ is a compound function; and ^ ~ | ^V" \, this being the
definition of the symbol | \f/ \j/" {: viz., \ W \ is a compound functional form.
[Examples in 2.04 (i), (ii), (iv); (vi) to (xii); (xiv).]
3.13. In general, \f/ f \f/ f ~ W \ is obviously false. The distinction
between compound and composite ^; is very important for the sequel. Clearly,
the \j/i might have been classified according to their factors when considered
as compound functions; a slight consideration is sufficient to convince that
1 Henceforth n shall always signify a positive integer; n is the symbol of any integer, not of
a particular integer. Also, \f/ t \f/ , etc.; ^i> are functional forms as denned in 0.06.
12 AN ARITHMETICAL THEORY
no appreciable progress can be made in regard to either (i) or (ii) of 0.00 in this
direction, and that such a classification is unnatural. The primitives defined
in 3.14 are fundamental in the theory of the \f/i, also for all that follows.
*3.14. Let (t) = ti, t z , , t r denote a set of positive non-zero integers,
no two of which are equal, and (t ) = ti, t 2 , , t/ any permutation of (t).
Also, let (h) = hi, h 2 , - , h r denote a set of functional forms abstracted from
factorable numerical functions hi(x), h 2 (x), -, h r (x) respectively [0.06; 0.04],
no one of which is a unit [3.10], but any one of which may reduce to a numerical
constant, and let a (1, 1) correspondence be established between (t) and (h) in
such a way that fa, hi are correspondents, and denote by (h ) = hi, h 2 f , , h/
that permutation of (h) in which hi corresponds to ti (i = 1, , r). Then,
if Pi T1 P 2 T2 P s r> is the resolution of n into its simple factors [1.02], a func
tion, H(n), of n may be defined unambiguously by the properties (i), (ii):
(i) H(ri) vanishes when any n (i = 1, , s) is not a member of (t);
hence in particular, H(n) = if s > r.
(ii) If every n is a member of (t), and if n =*= ti (i = 1, , s), then
H(n) = hi (Pi)h z (P 2 ) - h. (P 8 ). [Cf. also 5.14.]
The so-defined H(n) is a primitive function, or simply, a primitive.
f3.15. A primitive is a factorable numerical function. This follows at
once from the definitions in 3.14; 0.04
3.16. In order to specify completely a primitive, it is necessary to give
both the sets of integers and functions and the correspondence between them,
from which the primitive is derived. In 3.17 to 3.19, the notation is that of
3.14. The definitions in 3.18, 3.19 are again fundamental.
*3.17. H is a primitive form.
*3.18. Let (t"), = ti", t>>", , t r " denote that permutation of (t) which
is such that t r " > t r -i" > > t 2 " > ti" , and let hi" denote the corre
spondent of ti" (i = 1, ,!*). Then (t") is the index of the primitive form H,
written, /(#); and (h") is the base of H, written B(H); = V, h 2 ", - , h r ".
*3.19. The index I(H) and the base B(H) [3.18] taken together, and
written in the form &,{ \ , constitute the characteristic of H, written K(H)
J
for brevity.
|3.20. If K(H) is given, where H is a primitive form, the primitive H(ri)
is uniquely defined; obviously.
f3.21. (i) V(n; a, 6, c, 1) is a primitive function [2.01].
(ii) If h(x) = cx a , then K(V) = PI; [2.01; 3.19]; * being abstracted
from ty(n; a, b, c, I).
(iii) \l/(n-, a, 6); x(n] a, 6); f(w; a, b) are primitives [2.02].
(iv) The right-hand members of 2.04 (i), (ii), (vi); (ix) to (xiv), are primi
tives. Of these, (i), (ii) may be verified from the definitions; (iii) follows
from (i), and (iv) from (iii), referring to 2.02, 2.04. As all these are mere
illustrations, and are not used in the deduction of any subsequent theorems,
OF CERTAIN NUMERICAL FUNCTIONS. 13
until 8, the verification of (i), (ii) may be omitted if desired, until 5; 7,
when both are proved instantaneously.
*3.22. In the definition of primitives [3.14], there is an apparent narrowing
of the development by the restriction that the hi(x) shall be factorable.
This is now removed as follows: if h(x) is not factorable, and if P = Pipj - Pk
is a simple number, then h(P) shall mean h(pi)h(pj) h(pk). Clearly, in this
case, h(P) is purely symbolic. An important restriction is now imposed upon
the primitives.
*3.23. If the functions hi(x) used in defining a primitive form H [3.14;
3.17], are polynomials in x with positive, negative or zero integers as coeffi
cients, the case in which every coefficient is zero being excluded by 3.14, then
H(ri) is an algebraic primitive, and H is an algebraic primitive form; provided
the degrees of the polynomials be all finite (if this is not included in the defini
tion as usually given, of a polynomial).
*3.24. Referring to 3.14, r is the extent of H(n) and of H. The extent is
of subsequent importance in that the case in which r is infinite (essentially),
gives an analogue to the entire transcendental functions. [Cf. 5.30.]
*3.25. Let Hi (i = 1, , r) denote algebraic primitive forms. Then
the meaning of H ~ HiH 2 - -H r is known from 3.07; and if HI, H 2 , , H r
are identical, viz., if K(Hi), K(H 2 ), , K(H r ) are identical in all respects
[3.19], H is the rth power of Hi, written, H ~ Hf (i = 1, , r). Hence,
H^ Hj, - - - , H k being any primitives, HfHf - H k l is defined when s,t, , I,
are positive integers >0;ifs = =---=Z = 0, the value is defined to be
e, any unit [3.10].
Similarly, \f/\f/ \l/ (r factors) is defined to be \j/ r , for \f/ as in 0.03. The enun
ciations of theorems and their proofs, concerning Hi forms, will be found in
5, 6; and are deferred for reasons similar to those stated in 3.21. The
majority of factorable functions in current use will be shown to be of the form
Hi s - -H k l , where s, , I are positive or negative integers; the meaning of
Hi s for s negative is now defined similarly to the usual procedure in algebra:
*3.26. If H is an algebraic primitive form [3.23], a any positive non-zero
integer, any function H which satisfies the equivalence H a H ~ e, where e is
any unit [3.10], is a value of H~ a , and H f is equivalent to t/H a , where e/H a is the
ideal reciprocal of H a , written, H <*> e/H a .
If e is replaced in the foregoing by 1 (the absolute unit), then l/H a is
the pure reciprocal of H a ; written H ~ l/H a .
Also, similarly, from 3.25, for ^ as in 0.03, e/\f/ a and l/^ are defined [cf.
3.07; note (1)]; also e/\J/ a \j/ b - - , is also obviously defined.
3.27. It is of course evident that the functional forms symbolized by
e/H a , l/H a , e/\l/ a , l/\f/ a , do not necessarily exist. 1 In the following definitions,
1 That is, without proof. It is doubtless possible to derive existence theorems from the
definitions already given, but as this appears to be complicated, all such are postponed until
after the introduction of several new concepts in 4, 5, upon which all become practically
obvious. Also, by deferring the proofs, sufficient material is provided for more than one dual
of arithmetic [0.00 (i)].
14 AN ARITHMETICAL THEORY
the significance of the concepts defined is conditional upon the existence of
ideal and pure reciprocals; viz., if these do not exist, the definitions are to be
considered as meaningless. However, it is proved in 5, that both kinds of
reciprocals always exist, so that the theory which is being constructed ia
actual and not merely formal. These remarks apply also to 3.35, 3.36.
f3.28. Any product all of whose factors are units, is a unit. For, if
61, 2 are any units, ^(n)fi(d)ez(n/d) [cf. 3.01 (i) for notation] consists of v(ri)
[2.041 (v)] terms, each one of which vanishes if n > 1, for then either d or
n/d is > 1. Also, if n = 1, the sum is Z(i)ei(l)e 2 (l/l) = 1, [3.10]; hence eie*
is a unit. From this and 3.08 it follows that ; (i = 1, , r) being units,
a* (i = 1, , r) a positive integer, ei ai e 2 " 2 c r ar is a unit.
f3.29. If e is a unit, \f/ any functional form [general, as in 0.06], e\j/ ~ \f/~
As in 3.28; in 2( n )e(d)t(n/d), each term except that for which d = 1, vanishes;
viz., the sum reduces to \f/(ri). Hence, etc.
*3.30. If in any equivalence, either of \l/, -^ may replace the other (without
affecting the validity of the equivalence), then ^, ty are identically equivalent:
symbolically ^ ~. \J/ .
Identity and identical equivalence are distinct relations. A functional form
^ is identical only to $, and is identically equivalent to \p, obviously; but if
^ is identically equivalent to $ , it does not follow that $ is identical to ^
[cf. 3.31]. The distinction is of the greatest importance in connection with
0.00 (ii) ; by means of the concept of associate functions [3.32], uniformity is-
introduced into apparently heterogeneous masses of functions.
|3.31. ^ ^ $. [Notation and proof directly from 3.29; 3.30.]
*3.32. a (i = 1, , r, - - ) being units, ^ any functional form, the
identically equivalent functional forms e$ are associates of $.
f3.33. In respect to ideal multiplication a functional form is indistinguish
able from any of its associates [3.30 to 3.32].
Hence, if ^ ~ ^i^ 2 * -j/v is a resolution of \f/ into ideal factors, by 3.31, ^,-.
may be replaced by e t -^ (i = 1, , r), where e is any unit, and by 3.02 (iii) y
3.28, the result of such replacement is \J/ ~~ tty\^i- - -^ r where e is some unit,
and the right of this equivalence is indistinguishable for purposes of ideal
multiplication from fafa -\I/ r .
3.34. Henceforth, a functional form \l/, and its associates [3.32] when con-
sidered as ideal factors, will be considered as identical, and will be represented
by $. In this connection, cf. also 4.20; 5.21; 5.22. This procedure is in
entire analogy with that in the usual Theory of Numbers; e. g., in connection
with algebraic numbers. When it shall be necessary to distinguish a ^ from
its associates, the same may be readily done.
*3.35. Units may be factors of a product, and yet be not explicitly in
evidence, as, e. g., in the arithmetical definition of certain functions. If
\l/, f, V are any functional forms, no one of which is a unit, such that
^ ~ \l/ \l/", and if it be possible to resolve ^ , t" further into (ideal) factors,
^/ ^ \f/i\l/ 2 f ; ^" ~ ti ^z", in such a manner that no one of ^/, ^2 , ^i"> ^2"
OF CERTAIN NUMERICAL FUNCTIONS. 15
is a unit, but some one of the products fa fa", fa fa", fa fa", fa fa" is equiva
lent to a unit, e, then e is a ZatoZ unit in W .
Let e.^ tf iVi" be a latent unit in ^V"; hence (in accordance with 3.34)
\{/ ~ fa fa" , and in fa fa" the latent unit e is said to be expressed from ^V"*
Proceeding similarly with fa fa", let ^ ~ fa fa", in which a latent unit
(if there be one) has been expressed from fa fa", giving \l/afa"* -V ^s process
terminates, giving, for r finite, \p ~ fa fa", then r/vV/ is the expressed form
of ^V" or of A- The expressed form of W is that product identically equiva
lent to W, in which there is no latent unit.
It will be shown that the latent units in any product W may be expressed
by one "twist of the press" [cf. 5.23; 5.24], and this applies to the following
also.
From fa fa - -fa latent units are expressed by considering pairs of factors,
as above; then in the result, pairs of factors, and so on, until if the process
terminates, giving for s finite, fa fa- - fa ~ ti (s ^z M - /v (s) wherein no pair of
the factors ^ (8) (i = 1, , r) contains a latent unit, faWfaW* .^ r <> is the
expressed form of fa fa fa.
*3.36. A functional form \f/ whose expressed form [3.35] is divisible
[3.07] by only \l/ and units [3.10], is, if \f/ is distinct from a unit, a prime form;
and \l/(n) is a prime function.
If H is a primitive form, which is also a prime form, H is a prime primitive;
if H is in addition algebraic [3.23], then H is an algebraic prime primitive, etc.
3.37. With the exception of assigning a meaning to if/i** where fa, fa
are functional forms [this is done in 8.09 to 8.23], the definitions, etc., of 3
provide sufficient material for the accomplishment of 0.00 (i) and (ii). But,
in order to derive the properties of the \f/t with a minimum of calculation, and
also to make the processes and foundation sufficiently broad to support
several duals of arithmetic, the further consideration of the ^- and their
properties so far defined, is based upon the theory of sets, characteristics, and
generators, the first two of which, themselves have an arithmetical theory,
and these will be defined and investigated in 4; 5.
4. SETS.
*4.00. Let t{ (i = 1, , r) denote r positive, non-zero integers, no two
of which are equal; and let > tj when i > j. Then, the when arranged
in ascending order of magnitude, constitute an integral set, or, where there can
be no confusion, a set, denoted by | ti, t z , , t r | or, by | t \ r . The elements
of \t\ r are the t -; the degree is t r , and the extent, r. Sets will occasionally be
denoted by S, S , - , Si, , etc., where it is unnecessary to put the elements
in evidence.
f4.01. If H is a primitive form, I(H) is a set. 1 [3.17; 3.18.]
*4.02. Sets, S, S , are identical, S = S , if each element of either is an
element of the other. Non-identical sets, S, S , are distinct; S H= S .
1 As elsewhere in the paper, the statement of a theorem is alone given, when the proof is-
obvious or a simple consequence of the definitions.
16 AN ARITHMETICAL THEORY
f4.03. The number of distinct sets of finite degree, t r , is finite. The
number of distinct sets of finite extent, r, is infinite.
*4.04. Partition of sets is defined as follows: Let S = \ t \ r ; S = \ t \,
be such that ti > t r . Then [4.00], t/ > ti (i = 1, -, r; j = 1, -, s).
Hence | t l} t z , , t r , ti, t 2 , - , t, \ is a set, S". Then S, S f together constitute
a partition of S"; written S" = S + ; or *S + = S". Inversely; if
1 < s < r, then | ti, t 2 , - , t s and | t s+ i, s+2 , - - , t r together constitute a par
tition of \ti, tz, - -, t r |; written in either of the forms;
or,
| tij t 2 , , t s |
*4.05. Summation of sets is defined as follows: Let S = \ t | r ; S" = \ t" \,,
and let t\" , 1% ", , t k r " be all the distinct integers chosen from among
ti, tz, -,*/; h", J 2 ", , t s ", and let ti, t*, - , t k denote the */" (i = 1, , k)
arranged so that > tj when i > j. Then, | t k is the summation of | r
and | t" .; written, | * U = M Ir + M" I.J or, | r + | <" |. = i | 4 .
*4.06. The laws of partition and summation of sets together constitute
the laws of addition of sets.
4.07. It is scarcely necessary to remark that as a relation, = has distinct
meanings in 4.02; 4.04; 4.05. In 4.02, = is the sign of a certain relation
between sets; in 4.04, = and + together are the sign of a single relation be
tween three sets of a particular kind; in 4.05 = and + are together the sign
of a single relation between sets of a particular kind, and, in general, it is
obvious that (= and +) has distinct meanings, although similar, in 4.05;
4.04. Like remarks apply to some of the sections following; in no case is
there cause for confusion, and this repeated use of certain signs obviates the
introduction of new symbols.
f4.08. Addition of sets is commutative and associative.
*4.09. With the notations of 4.04; 4.05, respectively; S is the difference
of S" and S , S f is the difference of S" and S; each of | t | r , t" |. is the dif
ference of | t k and the other. The difference of Si and $ 2 is written Si S 2 .
Note that a pure negative, S, is not defined; but cf. 4.53.
*4.10. Clearly, S S is thus far without significance, it is denoted by
| |, and is defined to be the identity set. With respect to addition of sets,
| is to have the formal properties of in relation to addition of integers;
e. g., S | = S- | | =b | = , etc. [cf. 4.21], combining:
f4.11. With respect to addition (of sets) all sets form a commutative
group. The identity element in the group is | |.
*4.12. Let , e , , d, , denote quantities each one of which can
assume only the value or the value 1 ; and let /, // denote positive, non-zero
integers; then e Z/ + e %" represents three positive, non-zero integers, viz.,
ti, ti", ti + ti". Considering | t \ r , \ t" | s , let ti ", t z ", -, t k " f denote all the
distinct positive non-zero integers represented by c Z/ + e %" (i = 1, , r;
OF CERTAIN NUMERICAL FUNCTIONS. 17
j = 1, - , s), and let h, t z , - , t k denote the I" (i = 1, , A;) arranged in
ascending order of magnitude; then, the product of | t \ r , \t" \ 8 is t \ k ; and
| t \ r , \ t" | constitute a pair of factors of | t \ k ] in symbols, | t \ r \ t" | = | |*;
or, \t\ k = t \ r \ t" |.; and | | fc is factorable by each of | * | r , | *" | s . This
defines multiplication for sets. E. g.,
| 1, 2, 3, 4, 5 | ! 1, 2, 3, 4, 5, 6 | H 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 |.
| 2, 5 | | 2, 4, 6 = | 2, 4, 5, 6, 7, 8, 9, 11 |.
f4.13. Multiplication for sets is commutative.
f4.14. Multiplication for sets is associative.
For, (\t r a \ t" | 6 )| t " \ C) | t \ a (\ t" \ b I i " | c ) each has as elements all the
distinct non-zero values of eV + e %-" + e"V (i = 1, * , a; j = 1, , &;
fc = 1, , c).
f4.15. Multiplication for sets is distributive with respect to addition of
sets; viz., t \ a (\ f" |* +1 i " | c ) = | | *" | + |* | a I i 1 " |c- This follows
similarly to 4.14, on referring to 4.06.
f4.16. The degree of the product of two sets is the sum of the degrees
of the sets. For, | t \ r \ t" s contains as its greatest element, t r + t s "\ and
tr j t s " are respectively the greatest elements of | t \ r , \ t" \ s .
|4.17. The extent of the product of two sets is greater than the extent of
either of the two sets. For, | t \ r t" |, contains t r + t" which is in neither
of \t r , | i" | , also, it contains all members of either.
f4.18. No sets S, S (as defined in 4.00) exist such that S = SS [4.16].
*4.19. Assuming that | a\ y a 2 , -, a r \ obeys the laws of multiplication for
sets, and is such that, S being any set, S \ ai, a z , - - - , a r \ = S, the symbol
I ctij az, , a r \ is defined to be a unit set of extent r.
f4.20. Every element a (i = 1, , r) in a unit set of extent r, is
[4.16]. Hence [cf. 4.10], | S = S. With respect to multiplication for sets,
unit sets of different extents, are indistinguishable, and may be each replaced
by | |.
4.21. In 4.10, 4.19, 4.20, | |, | 0, 0, , | are not [by 4.00] sets; but there
is no contradiction in referring to them as unit sets, etc., regarding this as a
new term, etc. Unit sets of extent > 1, are not strictly necessary to 0.00 (ii);
they have been defined for reasons given in 4.64. Henceforth, with respect
to multiplication of sets, | | alone is taken as the identity element. Thus,
the present theory of sets has but one unit [cf. 0.01 (i); 4.64]. It is important
to note that:
f4.22. With respect to multiplication for sets, sets do not form a group.
For, in general, multiplication has not a unique inverse; that is, if S, S are
given sets, a set S" may be determined in several ways so that S = S S".
It will be sufficient to show that S" may be found in two ways to satisfy
| 2, 3, 4, 5, 6, 7, 8, 9, 10 | = | 3, 4, 5 | S". Here, S" = | 2, 4, 5 |, or | 2, 3, 4, 5 |;
as may be verified directly by multiplication.
*4.23. If a set S admits as factors only the single pair | |, S, then S is a
prime set, or simply a prime. A set that is not prime is composite.
18 AN ARITHMETICAL THEORY
f4.24. Primes exist. For, \t\iis prime [by reductio ad absurdum from
4.16 or 4.17].
f4.25. The number of primes is infinite [4.24; cf. also 4.62].
|4.26. A composite set of either finite degree or finite extent, provided
that in the latter case the degree is not infinite, is the product of distinct pairs
of sets in only a finite number of ways [from 4.16; 4.17].
f4.27. The product of two primes may be factorable by a prime distinct
from either of the two. It suffices to find a single example in which this is
true. Clearly, each of 1, 2 |, 4, 5 | is prime; their product is | 1, 2, 4, 5, 6, 7 |,
which is also the product of | 1 | and 1, 4, 5, 6 |; and | 1 is prime.
4.28. It is now required to devise a definition of divisibility for sets which
shall restore to the theory of sets the fundamental theorem of arithmetic,
stated in 0.00 (i), and which shall result in a unique quotient in every case,
Si -T- $2- The method adopted admits of wide extension and application, it
is clearly suggested by many well-known and important processes in arith
metic, particularly in the Theory of Algebraic Numbers. First, an analytical
definition is given, and then the processes of addition, subtraction, multiplica
tion, and division for sets are exhibited on a lattice of unit squares, as the
actual processes are best performed graphically. The lack of a short method
for resolving a set into its prime divisors, is but another point of resemblance
between the theory of sets and arithmetic. It will be seen when the geomet
rical representation is examined, that certain of the following definitions are
more inclusive than is necessary for the investigation of primitives [3.14], to
which sets are presently applied [5]; this is accounted for in 4.64, and in
any case there is less difficulty in devising a general concept of divisibility
than there is in evolving a definition to fit only a particular class of elements.
As always, the unit set, | |, is written in explicit form; viz., S, t r , - , etc.,
never denote | |.
*4.29. Let S = SiS/ (i, j = 1, ) denote all the resolutions of a com
posite S of finite degree into a pair of factors, S t , S/. By 4.16; 4.03 the num
ber of such distinct pairs of factors is finite. Among all pairs Si, S/ , S/, Si;
will be some in which the degree of Si is less than or equal to the degree of
either factor in any other pair, and all these Si of lowest degree, constitute the
minimal class of divisors with respect to S, denoted by [So].
|4.30. With the notation of 4.29; each member of [So] is prime. For,
if not, let S (0 = S W S be a composite member of []- Then S = S (k) (S W S/) ;
whence S (k \ of lower degree than S Q (i) [4.16], is a factor of S; hence [So] is not
the minimal class of divisors with respect to S since S (k) is not a member of
[So] , a contradiction.
f4.31. The class [So] contains only a finite number of members. By
4.03, since all members of [So], by 4.29 are of the same degree.
*4.32. Each member of [So] is a factor of S [4.29]. This will be expressed
by: S is divisible by [S ] , symbolically, S = [S ]S"; which may be regarded
as equivalent to S = So (i >Sj" (i, j = 1, ) wherein S W (i = 1, ) represents
successively each member of [So]. With this notation:
OF CERTAIN NUMERICAL FUNCTIONS. 19
|4.33. If S = [So]S", all the sets S/ f , in number finite, are of the same
degree [4.29; 4.31; 4.16]. Also,
*4.34. With respect to a particular S/ there is a minimal class of divisors;
let all such minimal classes be denoted by [/ ] (j = 1, ) Denning the
degree of [S/ ] as the degree of any one of its members, among all the [$/ ]
(j = i t . . .) there will be certain whose degree is equal to or less than the
degree of any [S/ ]. All the members of all these minimal classes of lowest
degree may be put in a single class, [$J; the minimal class of divisors of S
with respect to [Sol , and, obviously,
f4.35. Every [S ]i (i) , where $i () (i = 1, ) are the members of [Si],
is a factor of S. This will be expressed by;
*4.36. S is divisible by [S ][Si]; or, S = [S ][Si]S, where S may be = | |.
4.37. If in 4.36, S is different from | , the processes and definitions of
4.32 to 4.36 may be continued in an obvious way, until finally an S identical
with | | is reached, and S = [S ][Si]- - []; and
f4.38. g is finite [4.16]. Eachmember of [Si] , ^ i ^ g is a prime, and
if D[Si] is the degree [4.34] of [Si],
D[Si] m D[S ]; D[S Z ] ^ D[Si]; , D[S ] ^ D[S a .i].
Also, each [Si] contains only a finite number of sets.
*4.39. If each member of [Si] is a member of [Sj\, and if each mem
ber of [Sj] is a member of [Si], [Si] and [Sj] are identical; [Si] = [Sj]. If
[Si] = [S,] = [S 2 ] - - [S r ], then [SJ[S 2 ] - - - [S r ] is represented by [&] ,
and in this notation there is the resolution into prime divisors S = [$ t -] r< [>S/] >
[SkY k , of any composite set S of finite degree, each of the [Si], [Sj], -, [/SJ
being a prime divisor. The degree of $ is denoted by D(S); and clearly;
t4.40. DOS) - riD[Si] .+ r,-Z>[S,-] + + r*Z>[S ft ].
*4.41. If in the resolution in 4.39, from [Si], [Sj], , [Sk] there be selected
the respective classes of sets whose extent in each case is a minimum, the
resulting classes are called principal, and are denoted by {Si}, - - etc., viz.,
{Si} contains all members of [Si] whose extent is equal to or less than the
extent of any member of [Si], and S = { Si } r { Sj } r {Sk} r " is the principal
resolution of S into prime divisors. Note that these are not logical products of
the classes; such are considered in another connection [ 9].
4.42. # In an obvious sense that need not be dwelt upon, especially in view
of its analogies to certain parts of the Theory of Numbers that will at once
suggest themselves, the resolutions given respectively in 4.39; 4.41, are unique
factorizations of a given set. By the resolution of S, the principal resolution
will henceforth be meant.
*4.43. The divisors of S, are, in the notation of 4.41;
{&} r/ {Sy} r/ {S k }" where ^ r/ ^ r<; ^ r/ ^ r,-; ; ^ r k ^ r k ;
| | being the divisor when r/ = 0; r/ = 0; ; r k f = 0.
*4.44. Si is divisible by $ 2 if every divisor of $ 2 is a divisor of Si.
20 AN ARITHMETICAL THEORY
*4.45. By means of a few obvious changes in the statements, from 1.01
is derived a definition of simple sets; \ is considered as simple. Similarly,
the whole of 1 may be rewritten upon sets, instead of upon integers, as a
basis. In particular,
f4.46. Any set may be resolved into distinct simple sets in one way only.
More generally;
f4.47. To any theorem regarding simple numbers there is a unique corre
spondent in the theory of sets, obtained on replacing number throughout by set.
The interpretation of the results is another matter; it is related to the
problems of analysis situs mentioned below, but, not being essential to 0.00
(i) or (ii), is not further considered here. But (cf. also 4.61-3.),
f4.48. Four distinct arithmetical theories exist for sets. The four are
distinguishable from one another according to the respective definitions
adopted for divisibility, which may be as in 4.43, or directly from 4.39 in a
manner similar to 4.43, and in either case there are the alternatives of prime
or simple sets as a basis. For a more general theorem, cf. 6.37.
4.49. The four theories of 4.48 are all similar; more precisely, they are
simply isomorphic. 1 Although so like in appearance, the respective results
to which they give rise when applied to the primitives, are totally distinct in
kind. One only is carried further in this discussion; that in which divisibility
is as defined in 4.43. A very brief account of the lattice representation of the
chief concepts in 4.00 to 4.47 is now given. The necessary definitions have
been so cast that the proofs of the validity of the several processes in relation
to sets, are self-evident; details, and the enunciations of the many theorems
suggested by symmetry, etc., in the diagrams, may be left to the reader
[cf. Figs. 1, 2].
*4.50. Let m, n each assume in turn all positive integral values from to
co ; the points (m, n) lie in the positive quarter plane XOY; OX, OY being the
axes of coordinates, chosen rectangular for convenience; the (m, n) are the
lattice points. A set, | t r , is represented in either of two ways upon XOY:
(i) The X-representation; consisting of the half-lines [x ti} (i = 1, ,
r), a half-line being that segment of any straight line which lies wholly within
XOY. By within is always meant the interior of a region and the boundaries
of that region. In the diagram, the half lines {x = ti} are to be dotted, or
light. The lattice points lying on y = 0, and which do not lie to the right of
(t r , 0), are (m, 0), for m = 0, 1, , t r . Through each of the (m, 0) which
does not lie upon a half-line, {x = ti}, is drawn a full, or dark half-line, parallel
to x = 0. These are the dark lines of | t r >
(ii) The Y -representation, is obtained by rotating the X-representation
through an angle of 7r/2, and then through TT about OX. The X set is read
1 The precise distinction between theories as simply or multiply isomorphic, based upon
the correspondences between the fundamental relations connecting their respective elements, is
made in the second part. All theories of this first part are only simply isomorphic to each
other and to the arithmetic of integers; although, from 6.33 to 6.37 may be derived theories
multiply isomorphic to arithmetic.
OF CERTAIN NUMERICAL FUNCTIONS.
21
in the direction OX. In either representation, the totality of light and dark
lines clearly depicts \t\ r unambiguously, and the following is evident:
f4.51. Addition. The X representations of | t r , \ t" ! are made simul
taneously upon XOY. The coincidence of two half-lines is to be represented
by a light line except in the case when the two lines are both dark, when the
coincidence is to be dark; all lines of either set that coincide with no line
of the other are to be left as they are.
f4.52. Subtraction. To represent
The total result is t \ r +\t
when this exists [cf. 4.09].
Proceed as in 4.51, except that all coincidences are to be dark.
29
o
o
2^, o
o
SSo o
2*0 o
oo o
o
o o o
o o o o
o o o o
ZQo
o o o o o
/6Q o
o o o o o
O
o o o o o o
O O O O O O
o o o o o o
o o o o o o o
"o
o o o o o o o
O
o o o o o o
o o o o o o o
o o o o o o
,,,,,
, 7, 8,9,10,11,12,13,14,15,16,17,18,19,201
Fig. I, (Multiplication).
o o
o o o o
13,4.5,6,7,8,9,10,1 1,12,13, 15,6, 17, 18,21,22,23, 28,29W 3,4,7, 10, 12,
Fig. 2, (Division).
4.53. A pure negative, S, may be defined by extending 4.50 to include
all of the upper half plane, and S is the reflection in x = 0, of S. Also,
S S = | is the half-line {x = 0), etc., and the properties of pure nega
tives may be developed as in algebra. Pure negatives have not been con
sidered because, in relation to primitives, they have no significance [but, cf.
12].
*4.54. *(i) A lattice point which lies upon a dark line is a node (in fig. 1,
heavy dots). Nodes will sometimes be indicated by small circles; and a line
which passes through nodes only by the nodes on it; this in order to keep the
diagram simple. By convention, (0, 0) is always a node.
22 AN ARITHMETICAL THEORY
*(ii) The half-lines {x + y = k}, for k = 0, 1, 2, - , o>, are diagonals.
The diagonal [x + ?/ = 0} is clearly (0, 0). The kth diagonal is {x + y = k\.
*(iii) That segment of any line which lies within [cf. 4.50 (i)] a closed region
which is everywhere convex, is a sect; a diagonal sect is obviously defined for
a given closed region everywhere convex.
*(iv) A sect is proper or improper according as all the lattice points upon
it are not, or are, nodes.
*(v) The half-line {nx my = 0} (m, n, + integers), intersects every
diagonal sect in the rectangle whose vertices are (0, 0), (m, 0), (m, n), (0, n).
Starting from (0, 0) and proceeding along {nx my = 0} to (m, ri), the
diagonal sects are numbered successively, 0, 1, 2, , (m + n); and j, where
^ j ^ (m + ri), is the suffix of thejth diagonal sect, and is proper or improper
according as the diagonal sect is proper or improper.
f4.55. Multiplication. The ^-representation of | t \ r , and the F-repre-
sentation of | t" \ s are made simultaneously upon XOY (or, vice versa); and
t r = m; t s " = n. [Cf. 4.54 (v).] The product, | t \ r \ t" is obtained by
writing down successively the proper suffixes of the diagonal sects in order,
proceeding as in 4.54 (v). The proof is immediate from the definitions in 4.12;
4.54; noticing that on the jth diagonal sect the sum of the coordinates of
any lattice point is j. In Fig. 1 , the nodes only are shown on the representations.
4.56. Division. There are two cases: (i) where it is known that the set
82 is factorable by the set Si, it is required to to find all sets S such that
Sz = SSi. (ii) Where it is required to determine whether an S exists such
that, for Si, Sz given, 82 = SSi; in other words, to resolve S into its prime
divisors. These are in order of difficulty, but both are simple for actual
examples if the degree is low.
f4.57. Division; Case (i) [4.56]. (i) To find a value of | t g such that
| t \ r = | t" \ s t \ g , and (ii) To find all such values.
(i) Both the X- and F-representations of | t r are made simultaneously
upon XOY, and then everything except the diagonal sect (proper) {x -f y = t r }
and the improper diagonal sects obtained by surrounding every lattice point
on the join of (n, 0), (0, ri), ^ n < t r f and (n, 0), (0, n) nodes in the respective
representations, by a small circle, is erased. This may obviously be done in
one step, without erasure; see Fig. 2.
The diagram now consists of a right triangle (0, 0), (t/, 0), (0, t r ), crossed
by diagonal sects (represented by successions of small circles) parallel to
[x + y = t r }. Complete the rectangle of which (0, 0) and (t,", t r - t a ")
are opposite vertices. Through each node of the ^-representation of | t" \,
which (node) lies upon {y = 0} draw a dark sect within the rectangle, parallel
to {x = 0). Certain nodes of the improper diagonal sects will lie upon the
sect {x = is"}- Through each of these draw a dark sect, parallel to {y = 0)
(within the rectangle). Examine the rectangle to see if any node on the
diagonal sects of the right triangle does not lie upon one of the dark sects that
have been drawn parallel to the axes. Through every such node draw a dark
OF CERTAIN NUMERICAL FUNCTIONS. 23
sect parallel to {y = 0). Then, a value of \t\ g is found by reading off in
order from (0, 0) to (0, t/ t s ") on {x = 0), the lattice points which are not
nodes.
The proof will be evident on comparing the rectangle with that in 4.55;
also,
(ii) All possible values of | t \ g are obtained by drawing dark sects as in
(i) [after "Examine," etc.] in such a way that no new diagonal sect, all of
whose lattice points are nodes, shall be added to those already in the right triangle.
For, if any such new diagonal sect be added, the set | t \ r is changed by the
omission of an element, and if no such new diagonal sect be added, \t r is
still represented (on the right triangle). Cf. 4.55. Any way of drawing the
dark sects as just described, gives a value of | t \ g , read off as in (i) ; the distinct
values of | t \ g so obtained give the complete solution of the problem.
Obviously, if the degrees of the sets are large, the geometrical problem of find
ing all the 1 1 \ g becomes very difficult; cf. 4.59. It may be remarked that these
processes will only be thoroughly understood by the working out in detail of
a few examples, which should be constructed first by 4.55 ; the like applies
to:
|4.58. Division; Case (ii) [4.56]. To resolve | t \ r into its prime (set)
divisors. The right triangle is drawn as in 4.57, with its improper diagonal
sects, etc., and in addition, the node formed by the intersection of each dark
line with the sect {x -f- y = t r ] is indicated by a small circle round the corre
sponding lattice point. These may be called the vertical nodes, for a reason
that will appear presently. On {x + y = t r }, choose any lattice point as a
vertex, and complete the rectangle whose opposite vertex is (0, 0). Clearly,
if the chosen lattice point is a vertical node, the rectangle, for no system what
ever of dark lines drawn as in 4.57 (ii), can represent (as in 4.55) a pair of
factors of | t | r ; for, in this rectangle, the diagonal sect {x + y = t r \, viz., the
vertical node, is improper. Choose, therefore, that lattice point on {x + y = t r ]
which lies nearest the X-axis, and which is not a vertical node, and proceed as
just above. If there be more than one quotient, obtained as in 4.57 (ii),
proceed similarly with each; and so on. In this way 4.29 to 4.39 is performed
graphically; and obviously, the principal resolution [4.41] may be read off
in each case by inspection.
4.59. The whole of division, and resolution into prime factors, may be
performed in either half of the right triangle used, which is bisected by {y = x}.
This requires less paper, but more patience. Obviously, the essential geo*-
metrical character of a set is preserved if the lattice and its lines, diagonals,
etc., be distorted in any way, provided, however, that the points of inter
section of the various lines with any one of them are preserved in the same
order. The question of being able to draw in the diagonals as required in
4.57, 4.58, and of the number of ways in which this may be done, may clearly
be formulated in more than one way as a problem of analysis situs (or, pos
sibly, more properly, topology), of the same genus as some considered by
24 AN ARITHMETICAL THEORY
EuLER, 1 and whose complete solution depends upon the analytical theory of
sets. This may be developed by anyone who is interested; it is intimately
connected with the theory of sums of greatest integers, 2 viz., sums of expressions
[m/n], etc., and with the arithmetical integration, 2 mentioned in 8.25.
4.60. A theorem from the elements of the Theory of Assemblages is
used in the sequel so frequently that it may be stated for reference: Let
{A, A , A", -} denote a denumerable assemblage whose elements A, A f ,
A", are all either finite or denumerable assemblages; then, the assemblage
of the logical sum of the elements of A, A , A", , is denumerable.
f4.61. The assemblage of sets is denumerable. [From the definitions
relating to sets and 4.60.]
f4.62. The assemblage of primes (viz., prime sets) is denumerable. For,
first, this assemblage contains | t |i (t = 1, 2, , oo); second, it is a proper
part of the assemblage of sets.
f4.63. The assemblage of minimal classes of divisors is denumerable.
4.64. It may now be seen in what sense the sets constitute an arithmetical
field, and hence, in what respects there is an arithmetical theory for sets.
The theory is obviously an extended arithmetical theory in the same sense
that the theory of algebraic numbers is; for, the minimal classes of divisors
are not sets, that is, the prime elements are not in the original data of the theory,
regarding those data as sets, but the primes are classes of sets. This corre
sponds in an obvious manner to the introduction of ideals in connection with
the theory of algebraic numbers; for, it was shown [4.27] that an essential
property of arithmetical division is not valid for factors, but is restored on the
broader basis of classes, which in a definite way includes the special case of
factorization. Or, by a slight change ab initio in the point of view, an arith
metical field, and hence, theory, in the strict sense of 0.01 may be easily
imagined upon the sets as a basis, by considering each set as the limiting
case of a class of sets; viz., the limiting class is to contain but a single member.
This is in analogy to the principal ideals of arithmetic, in which, only by a
similar change in the usual meanings of the words (viz., in the arithmetic of
integers) can an integer be called a unique product of prime ideal factors. The
theory of sets is not considered further, except in its relations to the primitives-
It is now apparent that 4.00 to 4.63 is but the simplest case of a more general
theory of sets, in which the elements of a set are themselves sets, which in turn
is included in the case in which the elements of these set-elements are again
sets, and so on. In the general case, the lattice is in n-space. There Seems
to be no example at present in arithmetic of these more general numerical
functions corresponding to the n-space lattice.
*4.65. S = S mod S", is; "S - S exists and is divisible by S"."
1 EULER: Solutio problematis ad Geometriam situs pertinentis. Berol. Acad. Sci., 1759.
The writer has not had an opportunity of consulting this paper during the last two years, and
the reference is taken from LUCAS: Theorie des N ombres, p. 96.
2 The reader who wishes to pursue the subject may consult J. HACKS: Acta Mathematica,.
t. xvii (1893).
OF CERTAIN NUMERICAL FUNCTIONS. 25
4.66. There is a theory of congruences for sets; also a theory of forms;
but as these belong properly to the second part, when the congruence of
functions is considered, they are deferred until then. But it is clear that to
any S modulus there is a finite complete system of residues, etc.
5. CHAEACTERISTICS; GENERATORS.
*5.00. The base of a primitive has been defined, 3.18. More generally, if
it be possible to establish a (1, 1) correspondence between the positive, non
zero integers and the members of an assemblage of functional forms, /i, / 2 ,
, f r , - , in such a way that r and f r are correspondents, the concept of a
base may be extended, thus; let ti and // be correspondents, where ti is an
element of | t | r , and // is an // (j = 1, 2, , r, - ); then, the //, when
arranged in the order //, / 2 , , //, constitute the functional set, [/ ] r ; and
| t \ T and [f] r are corresponding sets, or simply, correspondents.
5.01. Throughout 5, H s denote primitive forms, [3.14; 3.17]; also,
cf. 4.01, 3.18, B(H) is [h] TJ I(H) is | t | r , [4.00], whence, for i = 1, - - -, r, the
elements hi, ti correspond. K is defined in 3.19. For 0.00 (i) and (ii), the
laws of combination of symbols [/] r , [5.00], presently given, are fundamental.
The introduction of generators is merely a convenience; all proofs may be
given without their aid, but at much greater length; here, the theorems
become self-evident. Also, cf. 3.22.
f5.02. Each of the sum, product, or difference, the last being as defined
in 4.09 but not as in 4.53, of two integral sets, being again an integral set is,
obviously the index of at least one primitive; directly from 3.14, 3.18, 4.01.
Clearly also, the functional sets corresponding [5.00] to each of these combina
tions of integral sets, as yet are arbitrary; they will therefore be defined in a
manner appropriate to ideal multiplication and addition, the last being wholly
distinct from algebraic addition [cf. 5.33].
*5.03. Let [f] r , \ t r be correspondents; likewise [/"] s , | t" . and [f] ,
\t g] also let the integral sets be such that | t \ r + \t" | = \t\ g \ then [f] 9
is defined 1 to be [/ ] r + [/"] s where, for i = 1, , g,
(i) fi = ft if ti is in | t \ r but not in | t" | 5 ;
(ii) fi = fi" if ti is in | t" , but not in ] t | P ;
(iii) /. = // + fi" if ti is in | t \ r and in j t" | 8 .
The possibilities are evidently exhausted, as, by 4.04, 4.05, ti must be either
in | t f \ r or in | t" s , or in both, and cannot be in neither. The result of adding
the functional sets may be written [cf. 4.07], [f] g = [f] r + [f"] s , or similarly,
with the members transposed, which implies, as for sets, that the latter also
is part of the definition; the latter is to apply to subsequent definitions and
need not be stated explicitly.
|5.04. Addition for functional sets is commutative and associative.
*5.05. If in 5.03 the sign of every // be changed, the result defines
[/ ]r - [/"].. This exists only if I t \ r - \ t" . is as in 4.09.
1 / / +/" is to be considered as abstracted from fi (n) +//(n), by an obvious extension of
" abstraction " as denned in 0.06.
26 AN ARITHMETICAL THEORY
*5.06. With the notation of 5.03, let all the solutions of Z/ + t" = ti
(1 ^ I ^ g) be given, for /, t/ respectively members of \t f r , \ t" and I
fixed, by t a + t," = fc; t b f + t y " = t t ; - ; t c + t z " = ^; and write [cf. 3.12]:
i //// i _ i f ff rf i i i / // // i _j_ i i f if n .
l/i I = Ija/x I 4- [/*/ | -r + \jcjz ;
and let a (1, 1) correspondence be established between the ti, fi as follows:
(i) fi = I //" I + fi if ti is m\t \, but not in | t" |.;
(ii) /, = | //" | + // if fc is in | r |. but not in t | r ;
(iii) /i = I //" |+ // + // if U is in | r and in *" | 8 ;
then, as in 5.03 the possibilities are exhausted, and by 5.00 the fi form a
functional set [/] defined to be [/ ] r [/"]., the product of [/ ] r , [/"].; or,
[/ ]r[/"]s = [f]g- This defines multiplication for functional sets; and,
|5.07. Multiplication for functional sets is commutative, associative and,
with respect to addition as defined in 5.03, distributive.
5.08. The proofs of 5.04, 5.07 may be seen immediately from the defini
tions, or intuitively from 5.13. Multiplication for sets is basic for ideal
multiplication [ 3], and prepares the way for establishing an isomorphism
between arithmetic (in toto), the arithmetical theories of this paper, and the
theory of polynomials in two variables, each with all; and more generally, on
the obvious extension of 5.06 is set up (in the second part), such an isomorphism
between the theory of rational algebraic functions and those mentioned in 4.64.
*5.09. Conversely to 5.06, if [/]</, [f ] r are given, and [/"], exists such
that (i): [/ ] r [/"] a = [/] then [/"], is a value of the quotient of[f] g by [f ] r .
In general, [f"] s is not unique, and it may be taken as the symbol of the entire
class of functional sets satisfying (i); and any relation involving [/"], is
considered to be significant only if the relation is valid for every member of the
class denoted by [f"] s .
*5.10. Referring to 5.01, 5.03, 5.05, 5.06, the right-hand members of (i)
to (iii) define the left:
\
> B(Hi)
\ r
=
f~-\ BVir \wiri rJ(#i) 7 (#2)l
(in) K(Hi)K(Hz) = \ I .
(iv) If X(#) exists such that K(H)K(Hi) = K(H 2 ), then #(#) is a value
of K(Hi)/K(Hi), etc. (as in 5.09). There is in (i) to (iii) the tacit assumption
that the right hand members are characteristics; this is true obviously from
the several definitions, the corresponding primitive being in any case uniquely
defined. If in (iv) no K(Hi) exists having the required property, viz., that
K(Ht)/K(Hi) be a characteristic, then K(H%) is prime. The further con
sideration of primitives, characteristics, etc., is much simplified by the intro
duction of
OF CERTAIN NUMERICAL FUNCTIONS. 27
*5.11. Generators. Let I(H) = \ t [ r ; B(H) = [h] r , and denote by z, x
independent variables; with H is associated the generator, T(H), where
T(H) = hi(x)z * + h z (x)z * + + hrWz *;
and 1 + T(H) is denoted by y(H). For all primitives, the independent
variables in the associated generator are to be x, z; y(H) is the generating
function of H.
f5.12. The difference of identical generators being excluded, it follows
from 5.11 and the definition of a primitive, that to each of the sum, difference
or product of two generators is associated a unique primitive; also,
t5.13. (i) If K(H!) + K(H 2 ) = K(H), then r(ffi) + r(# 2 ) = T(H).
(ii) If K(Hi) - K(H 2 ) = K(H), then T(Hi) - T(H 2 ) = T(H).
(iii) If K(H!)K(H Z ) = K(H), then y(Hdy(H*) = y(H).
(iv) The converses of (i), (ii), (iii).
All follow on comparing 5.00 to 5.10 with 5.11.
*5.14. Let now x = p, z = p~ s , where p denotes any positive prime
number, and s is as yet arbitrary (unity is not counted as a prime) ; with the
stipulation, that as variables, p, p~ s are still independent; e. g., if h(x) x,
t = 1, then h(x)z t is p/p s , and this is not further reducible, viz., is never l/p s ~ l .
The product sign extending to all prime numbers p,
is denoted by [y(H)], where I(H) = \t r , B(H) = [h] r . If [y(H)] be expanded
formally, the result will be denoted by [y(H)]^ the formal expansion, 1 ~ here
being, formally equivalent to. The evaluation of the formal expansions is
nowhere considered in this paper, so, strictly, questions of summability, etc.,
are irrelevant. However, it is presently shown that for the class of functions
discussed [cf. 0.00 (ii)], s may always be so chosen that all products and series
are absolutely convergent, and hence, = may replace ~. Formal multiplica
tions, etc., are to be carried out as if the series and products were already
proved to be absolutely convergent; in all cases the results have only a formal
significance.
t5.15. With the notation of 5.14, [y(H)] ~ Y,H(n)/n s . The case n = 1
is provided for in 0.03; the rest is by direct comparison of 5.14 (i) and 3.14;
cf. also 1.10, 3.22.
|5.16. $, \l/i, ^2 being functional forms abstracted from numerical func
tions as defined in 0.03 (N. B. Not necessarily factorable), connected by
the relation ^ ~ ^ 2 [3.01], then tiW/n s ] ^(n)/n s - t(n)ln .
For, evidently the coefficient of l/n s in the formal product on the left is
/d), the summation being over all divisors d of n; this is ifr(ri).
1 In this connection, the sign ~ may be interpreted, asymptotically equal to, in the sense of
POINCARE. [Cf. T. J. I A. BROMWICH, Introduction to Infinite Series, pp. 330-337.]
28 AN ARITHMETICAL THEORY
The theorem is valid if ^, \l/i, fa are all factorable, for factorable functions are
included in those defined in 0.03.
f5.17. If #~#i# 2 , then [7(^1)7(^2)] ~ [y(H)]. For, formally,
[7 (#1)7 (# 2 )] is [y(Hi)][y(H^)] on rearranging the factors; applying 5.15, 5.16,
the result follows.
f5.18. If H is an algebraic primitive form [3.23] whose index [3.18] is of
finite degree [4.00], a finite value of s may be assigned (in several ways) which
00
renders both [y(H)] and ^H(n)/n s absolutely convergent, and hence, for this
n=1
value of s, [5.15], [y(H)] = f^HW/n*.
nl
For, referring to 5.11, 5.14; (i) | y(H) | < 1 -f | Wp)/p "|; and each
il
hi(p) is a polynomial (of finite degree) with integral coefficients, hence for
hi(p) there is a positive integral constant (viz., independent of p), m^ such
that | hi(p) | ^ m l p 9 ^ where gt is also constant. Hence, the right of (i) is-
3 1 + Tl J mip J /p tiS I; (ii). But, t { > tj when i > j, hence p^ is the L. C. D.
of the fractions in (ii), which ^ 1 + j | pV^ m^ \ j /p M ; (iii). All
the mi being positive integers, the numerator of (iii) is ^ mp t+0 , where
m = miW 2 m r ; g = g\ + #2 + + g r ] and t = t r * (ti + ^ + + t r ) ;
the result being obtained by multiplication, etc. Hence, m, t + g being
positive integers, independent of p, | y(H) < 1 + mp t+a jp irS ; whence, if
s> {log m + (i + gf) log p}/t r log p, [which is, in effect, independent of p>
for log m/log p, may, for the inequality be taken = 0], mp t+g /p trS < 1; hence,
CO
for s so chosen, [y(H)], and hence ]P H(n)jn s , is absolutely convergent.
nl
Similarly, it is clear, that for e 4= 0, preassigned, s may be chosen so that,
(iv) mp t+a /p trS < el; also, since m, t -\- g, t r are positive integers, the left of
(iv) approaches as a limit as s approaches co . No attempt has been made
to find a lower limit for s, as the actual values of the products, etc., are irrel
evant to an arithmetical theory as outlined in 0.01.
f5.19. If for all values of s which are such that both series converge
absolutely, f"Hi(n)/n- = # t (n)M then Hi(n) = # 2 (n); the Hi, H* being
n=l n1
as in 5.18. Assume that for n^m, Hi(n) = H 2 (n). Then, from the equality
of the two series,
H,(m + 1) +JTH 1 (m + n)
m + n
In this, letting s become infinite, from 5.18, each term on either side, except
the first, vanishes; hence H\(m + 1) = Hz(m -f- 1). But, by 0.03, #i(l) =
# 2 (1); hence the induction is complete.
OF CERTAIN NUMERICAL FUNCTIONS. 29
5.20. The theorems in 5.18, 5.19 need not be explicitly used; all results
found by their use, in relation to numerical functions, may be easily verified
a posteriori; but their use is not only convenient, but essential to the rapid
finding of new relations. But, it is emphasized that the introduction of an
infinite process, which has no place in a pure arithmetical theory, 1 is not
essential to the proof of any result in the present theory. In order to give
independent proofs of the theorems, it is only necessary to substitute the
actual sums, etc., in place of ideal products, etc., in the equivalences, and
perform the necessary reductions by direct calculation, when, in each case,
an identity results.
f5.21. If H ~ HiH z , and c[y(H 1 )j(H 2 )} ~ [y(H)], then c = 1. This is
obvious but important in connection with units. Also, if [7 (#1)7 (#2)7 (#3)]
~ [y(H)] } with H ~ H\H, then y(H s ) is 1, and may be represented by
y(H^)/j(H^), where H is an arbitrary primitive form. But also, 7 (#3) may be
represented by F(x)/F(x), where F(x) is entirely arbitrary; this leads to the
conception, important for the present purposes, of
*5.22. Relevant Units. The discussion being restricted to the class of
all functions derived according to the several definitions, 3, from algebraic
primitives as defined in 3.23, those units, e if which arise from [7 (# t -)/7 (#;)],
and no others, wherein HI is an algebraic primitive, are relevant, and henceforth)
unit, when used in connection with functions derived as described from algebraic
primitives, shall signify relevant unit. The method in which relevant units
are generated is: let [7 (#)] ~ JT \fri(ri)/n"; and [ljy(H^] ~ V ti"(ri)/n s , then
n=\ n\
oo
[y(JSi)fy(Hi)] ~ ]T \f/(n)/n s , where \p ~ W\ but clearly, the left is 1 for all
values of s, identically; hence \f/(n) is a unit function, and \f/ is written ; or
*5.23. Positive and Negative Functions. Let Hi (i = 1, 2, ) denote
algebraic primitive forms; a f (i = 1, 2, ) positive non-zero integers; e s
units. All expressions of the form H\ ai H^ nz H a a> , say H, are positive
functions, all expressions of the form e/H (ideal, cf . 3.26) are negative functions.
Also, if H is a positive, H" a negative function, H H" is a mixed function; in all:
The entire theory might have been constructed on positive functions,
regarding negative and mixed functions as (ideal) fractions; this, to a certain
extent, is done in connection with addition, but the method adopted seems
better for 0.00 (ii) as the majority of the numerical functions in common use
are mixed, and from this point of view it seems advisable to regard H and e/H
as distinct functions according to their arithmetical definitions [0.05], rather
than to force all into the positive mould. Again, some of the negative func
tions, e. g., v, [2.041 (v)], are highly important, whereas their reciprocals,
(e. g., e/v), that is, the corresponding positives, are of little importance arith
metically; e. g., e/v does not seem to have arisen yet in the ordinary arithmetic.
1 Cf. G. B. MATHEWS: Theory of Numbers, Part I, p. 1.
30 AN ARITHMETICAL THEORY
fo.24. (i) If the G. C. D. of 7 (#i), 7(^2) is y(H), and if y(H 1 ) = y(H)
; 7(^2) = 7(ff)7(#2 ), then [7(#i)M# 2 )] ~ [ T (#i )/7(# 2 )].
(ii) If s be chosen so that all the series are absolutely convergent, and
if *V"~6, also [7(*OJ ~ I> (w)/n and [y(#T01 "^ E *"()M then
y(\l/ )y(\I/") = 1; hence, [1/7(# )1 ~ ]C#"( n )/ n *J viz -> tne generating function
of the reciprocal (ideal) of \f/ is the reciprocal of the generating function of \l/ f ;
for [l/7(^)] is formally l/[y ($ )], and by hypothesis, here = may replace ~
(in connection with generating functions) ; hence,
(iii) Hi, Fj being algebraic primitive forms, a i} b } - positive non-zero integers,
(i = 1, , r; j = 1, , k), the generating function of the mixed function
H^H^- - .Hr a /Fi b iF z b *- -Fk"" s= H, is
| y(H) = l7(^i)} ai {7(^2)} 2 ---{7(^r)rV{7^i)} 6l {7(^2)r 2 ---!7(n)}^
For, by repeated application of 5.17, the generating function of Hi ai is
{y(Hi)} ai ; whence the result follows immediately.
f5.25. From 5.22, 5.24 (or independently) it is easy to see, that if H is
an algebraic prime primitive [3.36], y(H) is irreducible, and conversely.
[This follows easily by reductio ad absurdum from 5.17, and the definitions
in 3.36, referring also to 3.34 and 5.22.]
Thus, the further study of positive, negative and mixed functions is reduced
to a comparison, so far as ideal multiplication is concerned, with the theory
of polynomials in two variables, whose coefficients are integers, and whose
absolute term in each case is unity. 1
The reader may prefer to omit the rest of 5, also 6, and pass at once to the
illustrations in 7, 8, which deal chiefly with numerical functions in common
use, and illustrate sufficiently the arithmetical character of the theory which
is constructed in 1-6. Questions of ideal congruence, also of ideal forms,
will have to be left, for lack of space, until the second part. In all cases where
infinite series are used, it will be assumed that s has been so chosen that the
series converge absolutely [cf. 8.07].
5.251. Generators and generating functions have been defined; occa
sionally, in reference to t(n), either term will be used to denote the series
whose nth term is \f/(n)/n 8 . and this will be denoted by IV, or y(s), , etc.;
in no case will there be cause for confusion [used chiefly in the illustrations].
f5.26. If H is abstracted from a mixed function, then y(H) is (i) ir
reducible, viz., an algebraic fraction in its lowest terms, or (ii) reducible, and
the G. C. D. of the numerator and denominator is the generating function of
all latent units [3.35] contained as ideal factors in H. [5.24, and the several
definitions, etc.]; similarly,
|5.27. If the generating function of either a positive or negative function
1 The Introduction to Higher Algebra, of M. BOCHER [New York, 1907], Ch. XVI, may be
consulted for the necessary theorems on reducibility, etc., in regard to polynomials, which
henceforth will be assumed without further notice. The domain of reducibility is [1J; other
domains enter when the (ideal) theory of forms in relation to mixed, etc., functions is considered.
OF CERTAIN NUMERICAL FUNCTIONS. 31
be resolved into its irreducible factors, say
= f(x, z) = {/i (z, *)}"{/ 2 (*, *)}" {f s (x, z)}",
where the r s are all either positive or negative integers, then each fi(x, z)
is the generating function of a prime primitive, Hi, and H ~ Hi ri H 2 r2 - H 8 r>
is the (obviously unique), resolution of H into prime factors (ideal).
|5.28. H, HI, H z being positive, negative or mixed functions, each of
(i), (ii), (iii) is formally equivalent to 1 each of the other two (at once, from
comparison of (i), (iii) with (ii) and 5.11);
(i) H-H.H,; (ii) K(H) = K(H^K(H,); y(H) = y(H^(H,).
The like is obviously false, if the functions are not precisely as specified, viz.,
if they are of the more general kind in 0.03; for, in that case, there is no appeal
to the theory of polynomials, etc.
f5.29. Every mixed function being of the form given in 5.24 (iii), by
successive applications of 4.60, or, as a corollary from the theory of poly
nomials, it is easy to show that: (i) the assemblage of mixed functions is
denumerable; whence, as a corollary, or independently; (ii) each of the
assemblages of positive or negative functions is denumerable; and hence,
that (iii) the assemblage of characteristics of algebraic primitives (all units
being relevant), is denumerable.
5.30. Considering H, a positive or negative functional form, and y(H)
as a function of z, [3.11], by 3.24, the extent is finite, viz., in y(H) there is but
a finite number of terms. But, if all definitions, etc., so far, remain unchanged,
except that the extent is essentially infinite, a new kind of numerical function,
a transcendental numerical function is defined. E. g., if y(H) = I z/2 +
2 2 /3 z 4 /4 + , H is transcendental. At present, it is sufficient to dis
tinguish ;
*(i) algebraic numerical functions; those whose generators are algebraic
functions of z ,
*(ii) transcendental numerical functions; those whose generators are
transcendental functions of z.
As (ii) is considered fully in the second part, it need only be pointed out,
that, as functions of their arguments, what are here called algebraic numerical
functions, are transcendental in the usual sense of the Theory of Functions;
but (i), (ii) makes a rational distinction between certain classes of functions.
There seems to be no example of (ii) at present in arithmetic (or in analysis) ;
it corresponds to the case of an essentially infinite index, and hence, to the
segregation of all integers into an infinite number of classes irreducible to a
finite number of classes. The properties of functions (ii), are (as will be
shown), arithmetically [0.01] identical with the analytical properties of tran
scendental functions (in the ordinary sense).
*5.31. If in 5.24, each primitive is algebraic in the sense of 5.30; viz., if
1 In the sense of mathematical logic.
32 AN ARITHMETICAL THEORY
for each Hi, 7 (Hi) is an algebraic function of z, x, then the mixed function there
defined is an algebraic mixed function; and, the totality of algebraic positives,
negatives [5.30], mixeds [5.31], constitute the assemblage of algebraic numerical
functions; briefly, A-functions.
f5.32. All H s denoting primitive A-functions: (i) B(Hi)B(H 2 ) =
(#i# 2 ); (ii) K(Hi)K(Ht) = K(HiH t ); (iii) these remain true if either or
both Hi, H be multiplied ideally by any units, (iv) If a value of K(Hi)/K(H z )
exists, it is unique, (v) K(H) may be written as a product K(Hi)K(H 2 ) -
K(Hr) wherein K(Ht) (i = 1, , r) is further irresoluble into factors, in one
way only, K(HH } and K(H) being considered identical if H is a unit. Sim
ilarly, for B (H).
The proofs are immediate consequences of the definitions and the corre
sponding theorems on polynomials. The (v) is false unless the primitives are
A-functions.
*5.33. Ideal Addition. All H s being as in 5.32; referring to 5.13, (i),
(ii), H is unique when HI, H% are given, and, the primitive A-function [5.31]
whose characteristic is K(Hi) + K(H^} is the ideal sum of the primitive A-
functions whose characteristics are K(Hi), K(H 2 ) respectively; from 5.13 (ii),
the ideal difference of two primitive A-functions is similarly defined.
It is clear that in no quantitative sense is an ideal sum or difference a sum
or difference; the ideal sum expresses a relation between functions that is
only remotely connected with their arguments.
f5.34. Ideal addition is associative and commutative; and, with respect
to ideal multiplication, is distributive. Also (an important consequence for
0.00 (i) and (ii)), the property of factorability [0.04] is invariant under ideal
addition and subtraction.
*5.35. There is another sepcies of addition, important for the sequel;
it may be called mixed addition, and corresponds to the compound multiplica
tion of 3.12. If i/>, faj fa are any functional forms, ^i(n) + fa(ri) is not a
factorable function, since, for n = 1, the value is 2, violating 0.04. But
!Lt(d){fa(n/d) + fa(n/d)} [cf. 3.00], is evidently V(n) + t"(n), where
(n)
\f/ f ~~ \J/fa\ i// ~ \l/fa- and hence, treating \J/i + fa as if it were a numerical
function (as defined in 0.04), the ideal product of ^ and fa + fa is $fa + \//fa.
Similarly for fa -f ^ 2 + + fa (r > 2); and, with this significance,
fa + ^2 + + fa is a mixed sum, denoted by | fa + ^ 2 + * * + fa \ ;
and it has just been seen that
t5.36. (i) ^ | fa + fa + -f fa | ~ ifa + *fa + + Wr !; also,
evidently, (ii) | fa + ^ 2 || ^\ + fa |, the product being taken ideally,
- fafa + fafa + fafa + fafa -
f5.37. If $1, fa are factorable, ^i + ^2 is in general not factorable. For,
if dv(ni, HZ) = 1, the necessary and sufficient condition that ^i + ^2 be
factorable, is, that for all pairs ni, n% satisfying dv(ni, n 2 ) = 1, shall (i):
I faM + faM || fa(n z ) + ^2(^2) | ~ | fa(nin 2 ) + fa(niHt) |; but, fa(nin z )
= ^1(^1)^2(^2); and fa(ninz) = ^2(^1)^2(^2); whence, from (i), the necessary
OF CERTAIN NUMERICAL FUNCTIONS. 3S
and sufficient condition becomes, | 1^1(^1)^2(^2) + ^2(^1)^1(^2) 1^0; which
is not in general (if ever) satisfied.
5.38. The theorem in 5.37 is that which necessitated the introduction of
ideal sums as defined; without ideal sums, a complete arithmetical theory of
A -functions is impossible. Ideal addition is also the basis, in the second part, of
the theories of congruences and forms for functions (ideal) ; it is also neces
sary to the more advanced parts of the theory, when, in order to complete
0.00 (i) an analogue for functions must be devised for DEDEKIND S theory
of Ideals. This latter may, ab initio, be used as the groundwork, and simi
lar theories to those of this part, be constructed, and finally, the whole of
DEDEKIND S theory is placed in (1, 1) self-correspondence by means of the
numerical functions proper to it. Again, for 0.00 (ii), the whole theory of
characteristics, indices, etc., could have been omitted, and it is not to be
inferred that by these means alone can 0.00 (i) be carried out in relation to
0.00 (ii); but, it is one of the simplest ways, and also, by choosing it, when
taken in its most general form, a complete body of interconnected theories,
each isomorphic to the other and to arithmetic in a manifold way, is obtained.
Ideal addition and its consequences are only of secondary importance in
relation to the integers ; in regard to the arithmetical properties of the functions
as such, it is of fundamental importance. It is easy to see, combining the
results of this section, and comparing with 0.01;
f5.39. There is an arithmetical theory of characteristics of A-functions;
moreover, in this theory, division is unique in the ordinary sense of the arith
metic of integers, and not dependent upon classes (as in the case of indices);
similarly
f5.40. For functional sets, each set being the base of an A -function, there
is an arithmetical theory; ideal addition for bases being defined thus:
B(H l )+B(H*)=B(H l +Hz) when and only when K(H 1 )+K(H,)=K(H 1 +H 2 ).
*5.41. Referring to 5.24, the divisors of the mixed function there defined,.
provided that each H, F is a prime primitive, are defined by Hf l H<f* - H r ar /
F^F^- - -PS*; where ^ < ^ a< (i = 1, , r), and ^ fa ^ 6, (j = 1,
, k): and, by convention, all unit divisors, e, correspond to the case
a,- = ft- = (i = 1, ...,r;j = 1, ,*).
If any H, F is not a prime primitive, the corresponding generating function
is resolved, according to 5.25, into its irreducible factors, and then 5.24 as
assumed in 5.41 results; whence, etc. Similarly, from 5.27, the divisors of
a positive or negative function are defined ; whence clearly,
|5.42. An A -function has only a finite number of distinct divisors;
distinct functions (as divisors) being non-identical, these latter being defined in
3.34.
f5.43. (i) By means of to 5, it may be seen that there is an arith
metical theory for A-functions, which include most (if not all) of the numerical
functions in common use. (ii) For convenience of comparison with the ordi-
34 AN ARITHMETICAL THEORY
nary arithmetic, some general theorems are placed together in 6; in a few
cases only is it necessary to give formal proofs.
6. GENERAL THEOREMS.
6.00. Prime shall be as defined in 3.36; unit as in 5.22; function, when
unqualified, any one of the kinds in 5.23. Cf. 5.43 (ii).
f6.01. The assemblages of (i) positive, (ii) negative, (iii) mixed, functions
are denumerable. [At once from 5.24; (iii); 5.27; 5.25 by repeated applica
tions of 4.60.]
f6.02. Every unit is a mixed function; no positive function, and no
negative function, can be a unit.
f6.03. There is a denumerable infinity of units that are distinct arith
metically [0.05] from each other.
From 6.01, 6.02; the units are distinct arithmetically, for obviously, if
HI, Hz, are either positive or negative functions which are distinct, then,
the arithmetical definitions of HI/ HI; H^H^, are distinct. [Examples in
7.09.]
f6.04. There is a denumerable infinity of positive, likewise of negative
functions, that are prime.
That there is an infinity of such, follows from the known theorem 1 that
if y(H p ) = (1 z p )/(l z), where p is a prime integer, y(H p ) is irreducible;
applying 5.25; 6.01, the theorem follows; hence,
f6.05. The assemblage of primes is denumerable.
f6.06. The assemblage of functions is denumerable. [Combining 6.01
(i) to (iii).]
f6.07. As ideal multipliers, the units are identical.
f6.08. If the absolute unit be taken as the identity element, so that e and
1/e are ideal reciprocals, the units form a group with respect to ideal multiplica
tion.
|6.09. All units are identically equivalent.
f6.10. The numerical values for any general argument of any function
and all its associates [3.32] are equal.
f6.ll. Any function may be resolved into its prime (function) factors in
one way only. [Numerous examples in 7.] [Directly from 5.21 to 5.26;
paying attention to 3.34.] Cf. 00.1 (2).
t6.12. If a prime function is a factor of a product of functions, it must
divide at least one of the factors (the product and division being both ideal).
Similarly to 6.11.
t6.13. Any function is ideally divisible by a finite number only of distinct
primes, and has only a finite number of ideal divisors [5.42].
t6.14. If a prime divides neither of two functions, it cannot divide their
product. [N. B. Cf. 5.23; remarks; otherwise the theorem has no meaning;
also 5.41.] Similarly to 6.11; and
1 Cf. G. B. MATHEWS: 1. c., p. 186.
OF CERTAIN NUMERICAL FUNCTIONS. 35
f6.15. The prime factors of a positive or negative function are respectively
all positive or negative functions; those of a mixed function may be either.
|6.16. Neither a positive nor a negative function can contain latent
units; a mixed function may, but not necessarily.
f6.17. There is an arithmetical theory for functions. [By combining
the theorems of this and of 5.] In this theory, there is an infinity of units.
f6.18. There is an arithmetical theory for positive functions; in this
theory there is but a single unit, the absolute unit, 1.
|6.19. There is not an arithmetical theory for negative functions. [The
break comes in the lack of an ideal addition which shall make the ideal sum
of two negatives a negative.]
f6.20. With respect to ideal multiplication the functions form a group.
f6.21. With respect to ideal addition the positive functions form a group,
the identity element being, by convention, 0. [The relations of ideal addition
to all the functions are not considered until the second part; they correspond
to the rational numbers.]
|6.22. With respect to ideal multiplication, neither the positive nor the
negative functions form a group; but each forms a semi-group. 1
f6.23. With respect to algebraic addition and ideal multiplication, mixed
sums [5.35] form a group; and e being the respective identity elements; and
f6.24. In similar respects, each of, positive, negative, functions form a
semi-group.
f6.25. There is a denumerable infinity of functions which are associates
of (or which are identically equivalent to) a given mixed function.
f6.26. The abstract operational laws of ordinary algebra are valid when
the operands are functional forms and + is algebraic (giving mixed sums),
X ideal; also when +, X are both ideal, provided, however, that in this
case 2 the functional forms be abstracted from only positive functions [from
5; replacing the functions by their generatiors, applying the corresponding
theorems on polynomials, and re-translating to functions, etc.], whence
f6.27. Any algebraic identity in which occur only sums and products,
or either alone, and hence also powers whose exponents are positive integers,
is true when sums are interpreted as mixed, products and powers being ideal.
f6.28. There is no analogue in functions to a complete system of residues
for a modulus, 2 if only mixed sums occur in the theory, but, among many
others ;
f6.29. The sums being mixed, powers ideal, n a prime integer, the
\l/i (i = 1, , r) functions as defined in 0.03,
(^i + ^2 + + W n = ti n + tz n + + tr n mod n]
1 As this term does not seem to be widely used, it is defined : A set of elements form a semi
group if they have the group property, etc.
2 Restriction removed in the second part, when all theorems of this are made general
for a field in which all four elementary operations are ideal, or in which 1, 2, or 3 are ideal,
the rest ordinary.
30 AN ARITHMETICAL THEORY
viz., regarding \^i n as a new function, distinct from \f/ i} similarly ^i + + \f/ r
a function, which qua function is independent of \pi (i = 1, , r), then for
every value of the argument of these new functions, the foregoing congruence
holds. [Similarly to 6.26.]
f6.30. If only mixed sums are admitted, there is no analogue in the theory
to FERMAT S theorem. 1 For, by 6.28 there is no concept of a complete system
of residues. In the case of positive functions, such may be readily imagined
in connection with 5.33.
*f6.31. If a, b are relatively prime, \f/i, fa functions which have no ideal
factor common, ^1 ~ ^ 2 6 introduces the concept of irrational functions; it
will be shown that their properties are wholly analogous to those of irrational
numbers, when these \J/i, fa t are compounded according to ideal addition,
multiplication, etc. There are no examples of these at present in arithmetic.
Their extents are all infinite, and they are distinct from the functions in
5.30 (ii).
6.32. This list may be indefinitely extended; enough has been given to
show the arithmetical character of the theory which has been constructed
for the functions considered. It has not been thought necessary to write
out formal proofs, as all are entirely elementary, and indeed, the theorems are
direct consequences of 3 to 5; details may, if desired, be supplied as sug
gested in 6.26. But the following are of an entirely different nature, and
indicate that the theory already constructed is but the first in an infinite
chain of theories, all abstractly identical to each other and to arithmetic, and
in a sense which may be easily imagined, each link in the chain implies all
that precede it. They differ from one another only in respect of the elements
upon which they are constructed; also, each is directly applicable to the
integers, but, as the chain is descended, the properties of the appropriate
functions in each link, become more and more complicated, until, in most cases,
even at the second link, they may safely be said to defy verbal definition,
although as functions, their properties are as simple as in 6.00 to 6.30.
f6.33. The elements in the theory up to 6.31, are the functions which
have grown naturally out of the unique factorization law (0.01) in ordinary
arithmetic, the arguments of the functions being integers; call these the ^ (1)
functions. Starting from 6.11, and the ^ (1) functions as arguments, 1, 3
(especially 3.14) may be rewritten, replacing therein number by function, prime
number by prime function. In this way are defined primitive ^ (2) functions,
prime i (2) functions, and 4 to 6, down to 6.31 are rewritten, a i (2) function
being the result of replacing in a i^ (1) each prime number by a prime i^ (1)
function, etc., as will be indicated on rewriting 1, 3. [Thus, e. g., if \j/ (l} (n)
= <p(ri), then, if $ ~ ^i ai ^ 2 a2 tr ar (ideal product), the corresponding \t/(ri)
is \K1 l/M (1 l/^ r )W, wherein the product is to be distributed,
simplified as far as possible algebraically (into a mixed sum or difference of
ideal products, etc.), and each function in the result is to have the argument n.}
In particular, there will be a theorem 6.11 for ^ (2) functions; and there will
OF CERTAIN NUMERICAL FUNCTIONS. 37
exist prime \ (2) functions, upon which, and the new 6.11, the process may be
repeated, giving ^ (3) functions (whose arguments are i^ (2) functions), and so on.
Hence any theorem regarding ^ (1) functions may be expanded (rather, un
folded) into an infinite sequence of theorems. [Examples of this 6.33 are not
considered until the second part, where also the inter-relations of the \f/ w ,
\b (2) , , are considered.]
6.34. This indicates one direction in which the theories of this paper may
be extended. Another consists in basing all upon the theory of Ideals of
DEDEKIND, instead of upon the integers; and others, more fundamentally
distinct from the present theory than any of these, will also be given in their
proper place. Another, simpler than any of these, arises as follows:
f6.34. The theorems of 6 (to 6.33) may be shown to be valid for func
tions based upon the resolution of an integer into simple (rather than prime)
factors [ 1], by replacing n by n [1.03], p t by Pi [1.01], and 5.14 (i) by [cL
1.10],
the II extending over all simple numbers, unity excluded; together with a
few simple and obvious corresponding changes elsewhere. Or, from another
point of view, the result is a direct consequence of 6.11.
In the illustrations, it is a simple exercise to change each theorem into its
correspondent upon this basis, and with a little care, the results may be
translated in accordance with 1.
|6.35. Finally, at any stage in 6.33, the prime or simple resolution may
be adopted. Thus, up to the functions ^ (r) there are 2 r theories constructed.
These differ in that, when finally unfolded down to ^ (1) functions, the prop
erties of integers which are relevant to each of the 2 r theories, are distinct.
But, each of the 2 r is of the same kind as that constructed for the ^ (1) functions,
abstractly.
|6.36. The following special cases of 6.27 are useful; (i) if ^ ^ \l/\fa t
then fa ~ fa \[/ where fa fa ~ e. Proving independently of 6.27, 7(1/0
= 7(1^1)7(^2), from the given condition [5.28]; also 7(^2 ) = 1/7(^2) [5.24
(ii)], and, 7(^1) = y($)/y(fa) = y(t)y(fa ), whence from 5.28, the result
follows, (ii) In the same way, if fa ~ fa, and fa ~ ^4, then $\fa ~ fafa ,
^1^4 ~ ^2^3; fa/ fa ~ fa/ fa] etc.
|6.37. Upon sets as a basis, instead of upon integers, the i^ (1) functions
[6.33] may be constructed, and the arithmetical theory of these is developed,
by a few changes in the meanings of the terms, parallel to that for i^ (1) func
tions. Also, on these set-numerical functions, the processes of 6.33; 6.35 may
be carried out. The interpretation of the results is made in accordance with
4.59.
6.38. A further generalization may be noted, although its nature will not
be apparent until after 8.09 to 8.23. It consists in replacing the integers r
as well as the t, of 3.14 by functional symbols ^, and repeating 6.33 to 6.35.
There result properties of the functions in relation to their arguments that
38 AN ARITHMETICAL THEORY
are of an order entirely different to any so-far considered. It will also be
proved that any \l/ (r} function [6.35] is a definite ^ (1) function (but not con
versely); and that the ^ (1) functions maybe distributed into classes of \f/ (r)
(r = 2, 3, ) functions in essentially only one way. In the i^ (2) functions
[e. g., in 6.33], 1 is the symbol of any relevant unit. In the ^ (r) functions,
the relevant units e (r) play the part of unity in the arithmetical definitions of
^ (1) functions.
7. SPECIAL THEOREMS AND ILLUSTRATIONS.
7.00. These are chiefly upon the consequences of ideal multiplication;
much more general theorems are given subsequently, also, other general
methods of deriving and proving such relations will be considered. The most
of those in this section are from LIOUVILLE S four articles, 1 wherein they are
stated. From the present point of view, there are two problems in connection
with any given function: (i) the resolution into prime factors [6.11]; (ii) the
relation to other numerical functions. Of (i) a single detailed example will
suffice [7.01]; the similar results in other cases, only are stated; and (ii) may
be systematically accomplished, either from (i) by 6.26, 6.27, or directly from
the generators; both methods are useful, and each suggests new functions
and relations in great profusion. With but little practice, the direct reading
of the arithmetical definition [0.05] from the generating function (or generator),
and vice-versa, become very simple matters; thus, e. g., in 7.05; 7.07, all
results may be derived from first principles as in 7.01, from the arithmetical
definitions; once they are stated however, they become obvious at a glance.
The example in 7.01 is more complicated than any in 7.05 or 7.07. For
definitions of symbols, cf. 2.041; 5; 3.12.
f7.01. To resolve H(ri) = <f>(D 2 (n))v(n/D2 (2) (ri)) into its prime factors.
Clearly, // is factorable [0.04]. The finding of y(H) is reduced, therefore, to
00
the summation and reduction of 1 + ^H(p a )z a ; where z = p~ s [5.24].
a=l
But, H(p) = 2;
=P a -
And,
Hence,
+ 2{zH
Whence, substituting for <p( ) its value, and writing
1 + ^p a z Za - = S = 1/(1 pz 2 );
1 Journal des Math., 2 Se>. (1857), t. 2. Sur quelques fonctions numeriques. The theorems
from LIOUVILLE are in 7.10 and a few in 7.12.
OF CERTAIN NUMERICAL FUNCTIONS.
39
= (S - z*S) + 2(zS - z*S) = (1 + 2) (1 +*)(!- 2)/(l - pz 2 );
viz. [cf. 5.24],
Let
7 (ffi) = (1 + 2); 7 (# 2 ) -: (!+); 7 (#s) = (1 - 2);
Then, the required resolution is H ~ HiH^H^H^. since each generator is
irreducible, and there are no latent units [3.35]; and by comparing with 2.04,
2.041, directly, H ~ p \ ^ \ \ n z v \ \ toi/2 |; thus H is the ideal product of the
four primes [3.36], /*, | M 2 1, I M 2 " I, I 2^1/2 1 [5.27].
It is to be noted that | MM ! is written | /z 2 1; || ju 2 \ v\\ as | juV |; etc.; viz.,
ju 2 , in | ju 2 i> |, is not an ideal square; so in all cases. As all the functions used
in 7 are specializations of ^ [ 2], this will now be examined in some detail;
powers, \f/ r , | \J/i\J/2 s , of functions are ideal, and e is any unit, ei, 2 , distinct
units [6.03]; again, cf. 3.12 and 2.04, 2.041 for the meanings of the symbols.
t7.02. Writing ^ (n; a, b, c, 1) = ^/, for I = 1; and comparing with
2.01; 3.25; 3.26:
= (1 + cp a z b ) lm ; m any integer.
p a z b ):
(i) y(^i) = (1 + cp a z b ) 1 (ii)
Hence [cf. 2.02],
tT.03.
f7.04.
f7.05.
xx
-i 7 " [7.03, 5.22, 5.24].
(1)
(3) 7 (
(5) 7 (|
(7) 7 (|
(9) 7 (
(11) 7 (|
(13) 7 (|
f7.06.
primes [5.
f7.07.
(1) 7
(3) 7 (cr)
(5) 7 (|
(7) 7 (|
(8) ai< >
(9) 2 W
(10) iSKn
= (1 - z).
- p z).
+ p
(2) 7 (Av) =
(4) 7 (tzr) =
(6) 7 (
(8) 7 (
(10) 7 (
(12) 7 (| nur |) = (1- p 2).
(14) 7 (| V?U r |) = (1 + p 2).
+ P2).
22).
pz).
The fourteen functions whose generators are given in 7.05, are all
27], except (2).
- P 2).
(2)
(4)
(6)
- p2)
- Z).
- pz) 2 .
(r -
= v(n)v(n*); 7 ( 2 ) =
(2r -
- z).
40 AN ARITHMETICAL THEORY
(11) 2 (n) s ^(/> 2 ((n)); 7(^2) = (1 - pz 2 )(l + z)/(l + pz)(l - pz).
(12) 8 (n) s n/Z) 2 < 2 >(rc); 7 (/3 3 ) = (1 + pz)/(l -f )(!- 2).
(13) 4 (n) = *>(/Mn)); 7(184) = (1 + z) 2 (l - z)/(l - pz 2 ).
(14) 5 (n) s ?(Z> 2 (n)Mw/02 (2) (w)); T (fr>) = (1 + 2-0(1 + z)(l - z)/(l - pz*).
(15) 6 (n) s Z>,<(n); 7(18.) = (1 + z)/(l + pz)(l - pz).
(16) 7 (n) s D,( n ); 7(187) = (1 + z)/(l - pz 2 ).
(17) ft(n) s Kw/Z>2 (2) W); TdSs) = (1 + 2z)/(l +z)(l-z).
(18) 9 (w) s 0(Z) 2 (n)); 7 (W = (1 + z 2 )/(l - z); 7(^(1, 2)) = (1 + z 2 ); f(l, 2)
written for f(w; 1, 2).
(19) 10 (n) E= e(n/D 2 (n)); 7 (0 10 ) = (1 + z)/(l - z); or, /3 10 ~ ^.
(20) u (n) = d(n/D^(n)) } 7(^11) = (1 + 2)/(l + z)(l - z); or,/?n ~ /3 8 .
(21) /3 12 (n) = TD-(n/Z) 2 ( 2 >(7i)); faZL-er.
(22) T (| tiro- |) = 1/(1 + z)(l + pz). (23) 7 (| UK, \) = 1/(1 - pz)(l - p 2 z).
(24) 7(1 |) -(!+ (2 r - Dz)/(l - z).
(25) 7(1 -W !) = (1 + (r - l)z)/(l + z) 2 .
(26) ^ = Wo 7(W = V(l ~ s)(l - P r z). fr(n) = the sum of the rth
powers of the divisors of n; obviously, v.
(27) T (l !) = 1/(1 + z) 2 . (28) 7(1 Wr* !) = 1/(1 - P r z) 2 .
(29) 7(| U r $ r I)
(30) 7(1 Ur?.r \)
(31) 7(1 Wrf. I) = 1/(1 - p r+8 z)(l - p r z).
(32) T (l mi (w) |)-(1+ (2m - l)z)/(l - z) 3 .
These all are modifications of 7.03, which comes directly from 7.02; thus, all
the functions of 7.07, as well as all those of 7.05, arise from ^ [7.02], which
obviously gives co more, primes and composites. Enough have been written
down to illustrate the richness of this single primitive, the simplest of all.
From 7.05; 7.07 there are the following resolutions into prime factors:
f7.08. (1) *>~wo 2 ; (2) <p~nui; (3) <r ~ u*ui , (4) 0~ /z 2 \ U Q , (5)
\mO\~ JU-GT; (6) \Uiv\~ Mi 2 ; (7) \*\~ M 2 1 W; (8) a{ r > - | ^(r - 1)" | u<? ,
(9) W r) - I M 2 (2r - 1)" ! wo 8 ; (10) j8, ~ | M 2 H ^w ; (11) /3 2 - Wi | ID-MI || 1*1/^2^1/2 1
| M 2 I; (12) /3 8 - ^^o | ^iM 2 I; (13) 184 - ! M 2 2 1 fewi/a | /*; (14) l8 6 ; cf. 7.01; (15)
j8 6 ~ D 2 ( 2 > - ! M 2 1 I TD-M! | u^ (16) ft ~ D 2 - | M 2 1 I te/ 2 1 ; (17) & - | fv\ i*ru Q ;
(18) /3 9 ~r(l, 2)w ; (19) j8 10 ^^; cf. 7.08 (4); (20) /3 n - fa (21) /3 12 - m;
(22) | wo- - -nr | wwi |; (23) | MKT | ~ u^u^ (24) | O r \ ~ \ M 2 (2 r - 1)" | w ; (25)
| TzW r) | - f(r - 1, l)tD- 2 [cf. 7.07 (18)]; (26) cf. 7.07 (26); (27) | vw - -nr 2 ;
(28) \urv\~u*; (29) | w r f r | ~ | -nrw r | w r 2 ; (30) | w r r.r | ~ WrW(.+i)r; (31)
I <a* I ^ ^r +8 w r ; (32) | mi<> | - r(2m - 1, l)i*o 8 ; cf. 7.07 (18).
From 7.05; 7.07, are written down the following units; the (ideal) product
of any number of units being again a unit, these may be multiplied together,
some or all, to furnish new units; those written are of use in subsequent
reductions by expression [3.35].
t7.09. (1) W (2) M&I; (3) M r | nu r |; (4) m \ M 2 1; (5) fv; (6) |
(7) | pui | MCT; (8) tzr M 0; (9) | Tzr Wl | | u^ |; (10) | fc 2M i/ 2 1 | nuifauv* |; (11)
OF CERTAIN NUMERICAL FUNCTIONS.
(12)
| /x 2 |
(19)
(23)
; (16)
(13) | v 2 1 TIT/I*; (14) ft
if* I w 2 w ; (17) ft
; (20) I u l
(24) !
! | w lM
1 I MMi I
I ; (21)
(25) | Ur
\ u lf i \ w, (15)
; (18) ft
| /uw r | ; (22)
j (26)
pU r
Each factor in these units is a prime, or the power of a prime; hence, in
an exact sense these are the resolutions into prime factors of the units [this
term does not strictly come under previous definitions]; and obviously, each
unit is a mixed function. The cumbersome j MI/ 2^2^1/2 1 will be denoted by r;
the arithmetical definition is evident on referring to 7.05 (9); 7.03; 2.05 (i).
In the following, the first and last equivalence of each chain [e. g., in \f/i ~ fa ~
~ fa this is fa ~ fa], is a theorem stated by LIOUVILLE; the intermediate
links come from 7.05 to 7.09 by obvious substitutions, etc.; each link [e. g.,
^i ~ fa , fa ~ fa , - ; ^i ~ ^3; *! etc.] is clearly a theorem of the same
kind. The numbering of the theorems 1.1, 1.2, , 2.7, -, etc., means,
e. g., 3.9, the 9th theorem in the third article of LIOUVILLE; the omitted
numbers are considered together in 7.121. Also, the number of links in any
chain may be continued indefinitely by multiplication by units; here, the aim
has been to reduce the links as far as possible consistently with clearness;
*s are units.
u\v \
v I
| M 2 (2r - 1)" j
2r - 1)" | i/o 4
[2.3; 2.4].
(2.6)
(2.10)
\ \P \
7.10
(1.1)
(1.3)
(1.5)
(1.7)
(2.1)
(2.2)
(2.3)
(2.5)
(2.7)
(2.9)
(2.11)
(2.13)
(2.14)
(3.1)
(3.2)
(3.4) | v 2
(3.5) | v 2
(3.6) ^ 2 (r) ~ /ttwi | M 2 (2r - 1)" |
(3.7) OatW - wo | M 2 1 I M 2 (2r - 1)
(3.8) i (2r) (7 ~ ! /x 2 (2r - 1)" | u Q 2 uoUi ~ wW r) . For omitted numbers, cf.
(3.16) U | r | w Mai (r) ai (r \ (3.17)
(3.18) | | F ~ Mai ^z/ 2 - Wo 3 | M 2 (2 r - 1)
| nrO |
| -nr^ | ~ |
HI [GAUSS, D. A., 39]. (1.2)
VU\. (1.4)
I ^ |. (1.6)
(2.4)
p<? ~
(2.8)
6 \ i (
(2.12)
M 2 ~
(2.15)
(3.3)
e.
- 1)
7.121.
42
AN ARITHMETICAL THEORY
(3.19) a,< 2r >0
(4.1) ft ~ W
(4.3) W r f.
(4.5) ! Wa i (2) ! f r ~
(4.7) UQ | W r f r ! ~
(4.8) U r Sr ~ U U r U 9
(4.9) I U m+n l-n | T
(4.10) I Wmf n+H T
(4.11) In (4.10) put
(4.13) | 6 n | Tm ~ w w
M 2 1
and
(4.2)
(4.4)
| r | i (2) . For 3 
