Mean values of multiplicative functions and applications to residue-class distribution

Abstract

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi-Erd\H{o}s function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo $q$, where $q$ varies uniformly up to a fixed power of $\log x$. We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the "correct" moduli $q$ that vary uniformly in a similar range, and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the "incorrect" moduli.