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

TL;DR

This paper establishes uniform bounds on the mean values of multiplicative functions, applies these to analyze the distribution of the sum-of-divisors function in residue classes, and extends previous results with optimal parameters.

Contribution

It provides new uniform bounds on multiplicative functions and applies them to extend distribution results of the sum-of-divisors function in residue classes, improving previous work.

Findings

Values of σ(n) are asymptotically equidistributed among coprime residue classes mod q for odd q.

Equidistribution for even q holds when restricting to n with many large prime divisors.

The results are uniform for q up to (log x)^K and improve existing bounds in the literature.

Abstract

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 \'Sliwa 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 $q$.