Mean values of multiplicative functions

Abstract

Let f(n) be a totally multiplicative function such that | ƒ (n)|⪯ 1 for all n, and let F(s) = ∑ ∞ n=1 ƒ(n)n —∞ be the associated Dirichlet series. A variant of Halász"s method is developed, by means of which estimates for ∑ N n=1 ƒ(n)/n are obtained in terms of the size of | F(s) | for s near 1 with ℜs >1. The result obtained has a number of consequences, particularly concerning the zeros of the partial sum U N (s) =∑ N n=1 n-s s of the series for the Riemann zeta function.