R. VAIDYANATHASWAMY (Madras - India)
THE THEORY OF MULTIPLICATIVE ARITHMETIC FUNCTIONS
Abstract (*).
I. - An arithmetic Function f(M±, M2,.... Mr), of r positive integral arguments,
is said to be a « Multiplicative » Function, if
fiMLNLi M2N2,...., MrNr)=f(ML,
M2,...., Mr) xf(^,
N2,.... Nr),....(l)
for all values of the arguments such that the products MiM2....Mr, NiN2....Nr
are relatively prime. Also, I call / a « linear » function if the equation (1)
holds for all values of the arguments. If f is a linear function, it is easy to
see that we can write; f(Miy..., Mr)=IIFk(Mk),
where Fk is the linear function
k
of one argument, defined by ;
Fk(Mk)=f(l,....,l,Mk,l,....l).
The two linear functions E, E0 are defined b y :
E(Mi,M2,.... Jf r ) = l for all values of ML, M2,.... Mr,
( 0 otherwise
)
These are fundamental, as well as the multiplicative non-linear function Ek (which
reduces to E, for k=l and to E0, for k=0) defined b y :
Ek(ML,....,
Mr)=k*;
v= number of distinct prime factors of MiM2....MT.
Multiplicative functions are well known, as instances of them occur frequently
in Arithmetic; but they have not been systematically studied in themselves.
The object of this paper is two-fold, to frame a complete calculus for the study
of these functions, and to establish a structural theory in which the linear
functions play the role of ultimate elements. The method used is a new one,
(*•) The paper in full is under publication in the transactions of the American Mathematical Society.
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which I have called « The Method of Generating Series ». The Multiplicative
property of f(ML, M2,.... Mr) implies that the values of f for all arguments can
be found by simple multiplication, if the values of f(pmi,p™*,....,pmr) are known,
for every prime p, and every set of indices mif m2,...., mr.
We call the latter set of values an element of the Multiplicative function f,
corresponding to the prime p. With each element of f, we associate the powerseries :
„
fp(x„x2,...., av) = 2 f(pmi,pm*,....,pmr)x^xt*....
^ .
mii...,mr
This series is a « Generating Series » of f, corresponding to the prime p.
Every multiplicative function must take the value 1 for simultaneous unit
values of its arguments ; hence every generating series has unity for its constant
term. If / is a linear function, each of its generating series is the expansion
of a rational function of the form (1— CiXi)~L (1 — c2x2)~~L.... (1 — crxr)~l.
The five processes of the calculus.
II. - The five processes to be described are all applicable to non-multiplicative functions, but their importance lies in the fact, that when performed on
multiplicative functions they yield only multiplicative functions.
These processes are:
1. Multiplication
of Functions. Symbol:
(fx <P)(ML, M2,.... Mr) =f(M,,.... Mr) x
®(ML,....Mr).
2. Convolution of
Arguments.
3. Composition of Functions. A <P denotes the composite function of f
and 0. P never denotes fxf, but always f>f. f>E is termed the integral (or
the numerical integral) of f.
4. Inversion of Functions. f^i denotes the inverse function of f.
5. Compounding of Functions. /*+ @ denotes the compound function of
/ and <P.
The first four of these processes are known, though (2) and (3) have not
received names.
The Arithmetical significance of the processes (3), (4), (5) is simple. The
composite is arithmetically defined b y :
(A <P)(J#i, M2
J f r ) - S f(òu Ò» ir) 0 fê, Ç,
g,
summed for all divisors di,ô2,....ôr of MiiM2,....Mr respectively.
The inverse f~L, of a function /, is the unique function determined from
f-i-f~E0.
R.
VAIDYANATHASWAMY :
The theory of multiplicative
arithmetic
107
The compound is arithmetically given b y :
(/+ $)(Mit M2 J f r ) - 2 M , à, , òr) <2> (^, ^,...,£),
where the summation is now for all divisors e5lf ò2,.... dr of ML, M2,
M
Mr respec-
M
tively, such that di is relatively prime to -~, d2 to -jA and so on.
The conjugate of a function is the unique function determined from:
f'+ Conj
f=E0.
These processes are equivalent to simple transformations of the generating
series. Thus, multiplication of f and <P amounts to the multiplication of each
term in every generating series fp(xL, x2,.... xr) by the coefficient of the corresponding term in <PP. Convolution of two or more arguments in a function
amounts to the identification of the corresponding variables in its Generating
Series. Composition of f and <P is equivalent to the multiplication of their Generating Series, while the inversion of f is equivalent to replacing each generating
series fp(xiy x2,.... xr) by the inverse series —
-. Lastly, compounding of
functions amounts to adding the corresponding series, and then replacing the
constant term by 1. The mutual relations of the processes (1), (2), (3), (4), (5), are:
1. Multiplication, composition, and compounding are each associative and
commutative.
2. Composition, has a restricted distributive property in respect of multiplication; viz., composition distributes multiplication, whenever the multiplier is
a « linear » function.
3. The Compounding Operation distributes multiplication unconditionally.
4. The Compounding Operation possesses also a quasidistributive property
in regard to composition, viz.,
f.(cpi+cp2+
.... + cpr) = (f. cpL)+ .... + ( / \ cPr) +
(Ei^rXf)i
where Ek(M\,.... Mr) is the function already defined.
5. Composition and inversion are permutable; i. e., the inverse of the
composite of number of functions is also the composite of their inverses.
6. The Compound of inverses of linear functions of one argument, is
equal to the inverse of their compound.
Theory of rational functions.
III. - A function f(N) of one argument will be called « A rational integral
function of degree n », if it can be expressed as the composite of n linear
functions.
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A function / of r(>l)
arguments will be called «An elementary rational
integral function of degree n », if it can be expressed as the composite of n
linear functions of the r arguments. By compounding the inverses of a sufficient
number of elementary rational integral functions of degree n, and taking the
inverse of the resulting compound, we obtain the general rational integral
function of degree n.
The composite P r « Q~L of a rational integral function Pr of degree r, and
the inverse of a rational integral function Qs of degree s, will be called a
rational function of degree (r, s). Also, Pr will be called the integral
component,
and Q8, the inverse component of the rational function.
A rational function of degree (1,1), will be called a Totient. The inverse
of a rational function of degree (r, s) is a rational function of degree (s, r) ; in
particutar the inverse of a totient in also a totient. More generally :
The five processes of our calculus are all rational processes in the sense,
that when performed on rational functions they produce only rational functions.
The systematic proof of this for functions of one argument, will be by
reducing the question to one of generating series, and utilising the theory of
partial fractions and recuring series. Also, the important fact appears in the
proof, that when we multiply, compose, or compound two rational functions of
one argument, the integral component K of the resulting function is related in
a very simple manner to the integral components JTi and K2 of the original
functions. When we compose or compound, K is simply the composite of JT*
and K2. For the case of multiplication, we write:
Ki=Pi'P2....
Pi,
E2 = Qi • Q2.... Qp,
PsXQt = Psti
where the functions P, Q and therefore also the functions R are linear. Then
K is the composite of the Ip linear functions R s t. It is not possible to give
an equally simple rule for the inverse component of the product., composite or
compound.
A rational function of one argument can be expressed in general as a compound of Totients.
Two cases of multiplication of functions of one argument are noteworthy and
have frequent applications. Firstly, it is easy to show that the product of two
Totiens is a Totient. If the Totients are P^P^
and Q^Q^, where P d , P 2
and Qi, Q2 are linear functions, then by the above theorem the integral component of the product is P i i = P 1 x Q 1 .
It is easy to show that the inverse component is given by :
where R^PsxQt.
W+*3+<*°\
(i^j-
R.
VAIDYANATHASWAMY :
The theory of multiplicative
arithmetic
109
The other case is that of the product of two integral quadratic functions of
one argument. For this, we have the result :
(P1.P2)x(Q1.Q2)=P11.P12.P21.P22.jr-i,
where K is an Integral Quadratic Function defined by :
(
( 0, if N ist not a Square,
\Pi(0)xP2(0)xQi(fN)xQ2(ÌN),
)=
)
if iVis a Square)"
Two examples of this formula may be mentioned:
1. Writing Pi=P2=Qi
= Q2=E, the theorem is equivalent to:
E2xE2=E2-E2.
This is the symbolic form of Liouville's formula:
St(d)e(f)-{T(jv)}«f
d/n
V
;
where x(N) is the number of divisors of N, and @(iV) the number of ways of
expressing N as the product of two relatively prime factors.
2. Writing Ia(N)=Na,
we have as a second example,
oa(N) x ah(N) = (Ia. E) x (Ih. E) =Ia+h. Ia. Ih. E- K'",
where :
. _ ( 0 when N is not a Square, )
{Ia+btfN) when N is a Square)
This proves Ramamijan's formula :
^ , aa(n) ab(n) __ Z(s)Z(s — a)Z(s — b)Z(s — a — b)
~
z(2s — b — a)
ns
2J
n
The Symbohc méthodes indicated here have numerous applications to the theorems
of arithmetic, practically all the functions that occur there being rational functions,
composed from certain definite types of elementary linear functions.
The Busche-Ramanujan identity for
oa(N)\
IV. - E. BUSCHE (*) proved that the function x(N)=E2(N)
of divisors of (N) satisfies the identity
or the number
T(WMV) = S T ( ? ) »
summed for the common divisors ò of u and v.
(i) Vide DICKSON, History of the Theory of Numbers, Vol. I, Chapter on sum and
number of divisors.
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Ramanujam noticed that a similar formula was also valid for oa(N), the
sum of the ath powers of the divisors of N, and utilised the inverse form of
the relation; viz.,
Oa(uv) = 2 * a (j) Oa (£} Ôaju(ô).
The existence of this identity is related to the following general problem:
II / be a multiplicative function of r arguments, then
f(uuf, vv',....)=f(u, v,....)xf(ur, v',....),
whenever the products uv...., u'v'...., are relatively prime; if these products are
not relatively prime, what is the relation between the two functions of 2r
arguments,
f(uu', vv',....,) and f(u, v,....) xf(u', v',....) ?
Cardinal and principal functions of two sets of arguments :
Let F be a function of two sets of r arguments (u, v,....), (u', v',....). On
putting u/=v'= .... = 1 in P w e obtain a multiplicative function FL(u, v,....) of r
arguments; similarly on putting u=v=.... = 1 in F, we obtain another function
F2(u', v',....). We call Pi and P 2 the two dérivâtes of F. The dérivâtes have an
invariant property given by the theorem :
The dérivâtes of functions of two sets of r arguments combine with one
another along with their parent functions in multiplication, composition, or
compounding. In other words, the derivate of the product, composite or the
compound of F and <P, is respectively the product, composite, compound of the
corresponding dérivâtes of F and <P.
Definition : A function of two sets of arguments will be called a Cardinal
Function, if each of its dérivâtes is the function E0. As an alternative definition, a Cardinal Function is one which vanishes whenever the products of its
two sets of arguments are relatively prime.
Also if F be an arbitrary function of two sets of r arguments, we shall
call the cardinal function which is equal to F when the products of the two
sets of arguments are not relatively prime, the Cardinal Function of F, and
denote it by Crd F.
It follows that the inverse of a cardinal function is also a cardinal function.
The following theorem corcerning Cardinal Functions is fundamental:
If F, Fr be two functions of two sets of r arguments, and if F=F' whenever the products of the two sets of arguments are relatively prime, then :
1. F is the composite of Ff with a Cardinal Function Ci,
2. F is the product of F' by the integral of a Cardinal Function C2,
3. F is the compound of F' and a Cardinal Function C3.
R.
VAIDYANATHASWAMY :
The theory of multiplicative
arithmetic
111
The proof is immediate by the application of the theorem on Derivates.
Now if f is any multiplicative function of r arguments, then from the multiplicative property,
, .
, v
M
n/.,
F
F
J
'
f(uu', vv',....)=f(u, v,....) x f(u', v',....),
whenever the products uv...., u'v'.... are relatively prime. Hence in the above
theorem we can take:
F=f(uu', vv',....)
F'=f(u, v,....)xf(u', v',....).
With this supposition, the functions C2, C3 are not capable of simple symbolic expression in terms of f; but it is found that d is the conjugate of the
cardinal function of f~~L(uu', vv',....), where f~L is the inverse function of /.
This evaluation of d gives the general answer to the question raised by
the Busche-Ramanujan Identity. To derive this identity itself from the value
of d, we require the concept of « Principal Function ».
Consider two sets of r arguments u, v...., u', v'.... in which each argument of
one set is associated with one of the other set; e. g. u with u', v with u', etc.
A function of the two sets of arguments is said to be a principal
function,
if it vanishes whenever two associated arguments are unequal. An interesting
property of the principal function is given by the Theorem :
The necessary and sufficient condition that a function of the 2r arguments
may be a function of the r g. c. d' s only, of the r associated pairs of arguments, is that the function be the integral of a principal function.
Now the condition that an identity of the Busche-Ramanujan type holds
for /", is that Ci = Conj. Crd f~L(uu', vv',....) be a principal function. It is shown
without difficulty that this can only happen when f is an integral quadratic
function. We have therefore the result :
We have an identity of the form :
f(uu>, wv-j-s f{l, I,....) r$,$r-)Wi, à,,-)
where the summation is for common divisors dL of uu', ô2 of v, v', etc., only
when f is an integral quadratic function.
When f is an integral quadratic function of one argument, we can determine the function F in the above identity immediately, Viz., if f is the composite of linear functions P 4 , P 2 , F(ô)=Pi(o)P2(ô)ju(ô).
This gives the following
extension of the Busche-Ramanujan Identity for any integral quadratic function
of one argument.
Any integral quadratic function f(u) = (Pi'P2)(u) satisfies the identity:
f(uv)=sKïMï)p^
p
«<*> /**),
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or in inverse form,
/(«M»)-2/$)Pi<A)p,(a),
the summation being for common divisors ô of u, v.
Äs regards the Cardinal Functions C2, C3, we have the following results:
1. C2 can be a principal function only when f is a totient.
2. C3 can be a principal function only if / is a totient or a rational
function of degree (2, 2).
As a verification of (1) we may observe that Euler's ^-function which is a
totient according to our definition, satisfies the relation:
x
${M) ${N)
— function of g. c. d. of M, N, only.