Multiplicative functions in short arithmetic progressions

Abstract

We study for bounded multiplicative functions $f$ sums of the form \begin{align*} \sum_{\substack{n\leq x \atop n\equiv a\pmod q}}f(n), \end{align*} establishing that their variance over residue classes $a \pmod q$ is small as soon as $q=o(x)$, for almost all moduli $q$, with a nearly power-saving exceptional set of $q$. This improves and generalizes previous results of Hooley on Barban-Davenport-Halberstam-type theorems for such $f$, and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well-known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli $q$ in the cases where $q$ is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every $q$. These results are special cases of a "hybrid result" that works for sums of $f$ over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matom\"aki-Radziwill theorem on multiplicative functions in short intervals. We also consider the maximal deviation of $f$ over all residue classes $a\pmod q$ for $q\leq x^{1/2-\varepsilon}$, and show that it is small for "smooth-supported" $f$, again apart from a nearly power-saving set of exceptional $q$, thus providing a smaller exceptional set than what follows from Bombieri-Vinogradov-type theorems. As an application of our methods, we consider Linnik-type problems for products of exactly three primes, and in particular prove results relating to a ternary version of a conjecture of Erd\H{o}s on representing every element of the multiplicative group $\mathbb{Z}_p^{\times}$ as the product of two primes less than $p$.