Multiplicative Functions with First and Second Means

Abstract

In Chapter six we proved, amongst other things, the following elegant result of Delange: Let g(n) be a complex-valued multiplicative arithmetic function for which |g(n)| < 1, (n = 1, 2,…). Then there is a non-zero mean-value

$$ {\text{A = }}\mathop {lim}\limits_{n \to 8} {\text{ }}n^{ - 1} \sum\limits_{m = 1}^n {g(m)} $$

if and only if the series

$$ \sum {p^{ - 1} (g(p) - 1)}$$

taken over all the primes p is convergent, and for at least one positive integer k, g(2^k)≠ − 1. It is important that the limit A is assumed non-zero.