Mean values of multiplicative functions

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