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

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 x for which the Alladi-Erdos function A (n) = ₏㵮 ₍ k p takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of x. We establish a similar result for the equidistribution of the Euler totient function (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 (n) among the coprime residue classes to the "incorrect" moduli.