On multiplicative functions that are small on average and zero-free regions for the Riemann zeta function

Abstract

In this short note, we prove the following result: If a completely multiplicative function f : ℕ → [−1, 1] is small on average in the sense that ∑_n ≤ xf(n) ≪ x^1 − δ for some δ > 0, and if the Dirichlet series of f, say F(s), is such that F(1) = 0, then for any ϵ > 0, ∑_p ≤ x(1 + f(p)) log p ≪ x^{1 − δ + ϵ}. Moreover, a necessary condition for the existence of such f is that the Riemann zeta function ζ(s) has no zeros in the half-plane Re(s) > 1 − δ.