Multiplicative functions that are close to their mean

Abstract

We introduce a simple approach to study partial sums of multiplicative functions which are close to their mean value. As a first application, we show that a completely multiplicative function f :with c = 0 if and only if f (p) = 1 for all but finitely many primes and |f (p)| < 1 for the remaining primes. This answers a question of Imre Ruzsa.For the case c = 0, we show, under the additional hypothesis p:|f (p)|<1 1/p < ∞, that f has bounded partial sums if and only if f (p) = χ(p)p it for some non-principal Dirichlet character χ modulo q and t ∈ R except on a finite set of primes that contains the primes dividing q, wherein |f (p)| < 1. This essentially resolves another problem of Ruzsa and generalizes previous work of the first and the second author on Chudakov's conjecture.We also consider some other applications, which include a proof of a recent conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions.