Mean values of multiplicative functions and applications to the distribution of the sum of divisors

We provide uniform bounds on mean values of multiplicative functions under very general hypotheses, detecting certain power savings missed in known results in the literature. As an application, we study the distribution of the sum-of-divisors function \(\sigma(n)\) in coprime residue classes to moduli \(q \le (\log x)^K\), obtaining extensions of results of Śliwa that are uniform in a wide range of \(q\) and optimal in various parameters. As a consequence of our results, we obtain that the values of \(\sigma(n)\) sampled over \(n \le x\) with \(\sigma(n)\) coprime to \(q\) are asymptotically equidistributed among the coprime residue classes mod \(q\), uniformly for odd \(q \le (\log x)^K\). On the other hand, if \(q\) is even, then equidistribution is restored provided we restrict to inputs \(n\) having sufficiently many prime divisors exceeding \(q\).