On the Nuinber of Solutions of F (n) --a for Additive Functions

Denote by r (n) the sum of divisors of n. A well known and probably hopeless problem in number theory states : Prove that a(n) = 2n has infinitely many solutions, i .e. there are infinitely many perfect numbers. More generally, one can try to estimate the number of solutions of a (n) /n = a, 1 < n < x. A method of Hornfeck and Wirsing [2] gives that for fixed a the number of solutions of a (n) /n = a, 1 < n < x, is o(x'). a (n) /n is multiplicative and the logarithm of a multiplicative function is additive. Henceforth in this paper we will study real valued additive functions. We will try to give upper bounds for the number of solutions of (1) f (n) = e, 1 <n<x. Denote by G (x, c) the number of solutions of (1). We will make various restrictions on f (n) in trying to get as sharp estimates as possible. To get non-trivial results we first of all have to exclude the ease f (n) = 0, henceforth this will always be assumed. First of all we prove the following simple TiaoRrnr 1. ror any f (n) we have uniformly in c G(x, e) < (1-Ef)x. To prove Theorem 1 observe that since f (n) is not identically 0 there is an integer iii. for wliieh f (m) ~ 0. In fact the smallest such m is always a power of a prime, m = poo. Let t be any integer with p,,-r t. Clearly f (t) 0 f (tpoo) and hence t and tpoo can not botli satisfy (1), or G (x, c)-~ x-Cpo° J ± Cpó-~1 < (1-el) x which completes the proof of Theorem 1 .