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FOURIER EXPANSIONS OF ARITHMETICAL FUNCTIONSBY ECKFORD COHEN
Communicated by Gerald B. Huff, October 5, 1960Let f(n), g(n)
denote complex-valued arithmetical functions with
( i ) ( ) - ) .d\n
W intner h as proved [3, 33] th at ifA gin)(2) n-i n
converges absolutely, then (w) is almost periodic (B) with the
absolutely convergent Fourier expansion,(3) f(n) ~ J^ arc(n, r ) ,
ar = ] >r - l n-l;r |n Wc(n, r) denoting Ramanujan's tr
igonometric sum. In addition, Wintner showed [3, 35] th a t if(4)
22is convergent, where r(n) denotes the number of divisors of n t
thenf(n) is represented for all n by its Fourier series,(5) f(n) =
^arc{n,r), ar = 2 3r1 n= -l;r|n W
In this announcement we point out that for certain
importantclasses of m ultiplicativ e functions (n ) for which (2)
is absolutely convergent (including many such examples that are
familiar), the convergence of (4) is not needed to ensure the
validity of (5). In fact,we have the following result.T HE OR E M
I. Suppose that f (n) is m ultiplicative and that (2) converges
abso lutely. In case either(i) g(n) is completely
multiplicative,or in case(iia) g(n) = 0 when n is not square-free,
and(iib) g(p)7 p for all primes p fthen (5) holds with the
convergence absolute.145
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1 4 6 ECKFORD COHEN [JanuaryAn analogue of this theorem for
functions of finite abelian groupsis proved in [2, Theo rems 6.1(a)
and 7.3]. W ith app ropria te changesin notation and terminology
the proofs carry over to the case of theintegers as expressed by
Theorem I.The above result can be reformulated in the following
simplemanner. Let 2 ' indicate summations restricted to square-free
integers and place
(6) h(n) = S ^ d\n dTHEOREM I I . Suppose that f {n)
ismultiplicative and that (2) converges absolutely with sum a. Then
if (i) is satisfied,
(7) /W=(- )c(,r);on the other hand, if (iia) and (iib) are
satisfied,(8) /W= "Z'(^W,'),r-i \rk(r)/th e convergence of both (7)
and (8) being absolute.PROOF. Let g(n) satisfy (iia) and (iib). The
analogue for integers of[2, Lemma 7.2] asserts that h(n)?Q for
square-free nt and tha t
* g(n) a(9) 2-f = ' r square-free.n-l ; (n ,r ) - l n A(f)By (1)
, gin) is multiplicative, and hence from Theorem I, (5) is
validwithA *() ~ g(ra) g(r) " g{a) ag(r)a r= 2, = 2, = ln=l;r|n ra
r o-i ;(a,r)-i a rh(r)
for square-free r, by vi rtu e of (iia) and (9). Evid ently , by
(iia) and (5),ar = 0 if r is no t square-free. T his proves (8). T
he proof of (7) is similarbut simpler, and is therefore omitted.Le
t u s now consider some special cases. Le t a(n) deno te the sum
ofthe divisors of , 0(w) the Euler ^-function, \p(n) the Dedekind
\//-function [3, p. 43 ], and $(n) the "square-totient" [ l , 6],
denotingthe number of a (mod n) such that (a y n) is asquare.
Correspondingto the cases, f(n)=a(n)/n, 4> {n)/n, *l/(n)/n, and
$(n)/n, we haveg{n) = l/w, p(ri)/n, /x 2(w)/, and \{n)/n,
respectively, where /x(w) denotes the Mbius function and \(n)
Liouville's function. Each of the
