Multiplicative functions in short arithmetic progressions.

We study for bounded multiplicative functions f$f$ sums of the form ∑n⩽xn≡a(modq)f(n),$$\begin{align*} \hspace*{7pc}\sum _{\substack{n\leqslant x\\ n\equiv a\ (\mathrm{mod}\ q)}}f(n), \end{align*}$$establishing that their variance over residue classes a(modq)$a \ (\mathrm{mod}\ q)$ is small as soon as q=o(x)$q=o(x)$, for almost all moduli q$q$, with a nearly power‐saving exceptional set of q$q$. This improves and generalizes previous results of Hooley on Barban–Davenport–Halberstam type theorems for such f$f$, 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$q$ in the cases where q$q$ is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every q$q$. These results are special cases of a "hybrid result" that we establish that works for sums of f$f$ over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki–Radziwiłł theorem on multiplicative functions in short intervals. We also consider the maximal deviation of f$f$ over all residue classes a(modq)$a\ (\mathrm{mod}\ q)$ in the square root range q⩽x1/2−ε$q\leqslant x^{1/2-\varepsilon }$, and show that it is small for "smooth‐supported" f$f$, again apart from a nearly power‐saving set of exceptional q$q$, 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 a ternary approximation to a conjecture of Erdős on representing every element of the multiplicative group Zp×$\mathbb {Z}_p^{\times }$ as the product of two primes less than p$p$. [ABSTRACT FROM AUTHOR]