Multiplicative functions in short arithmetic progressions

We study for bounded multiplicative functions f $f$ sums of the form ∑ n ⩽ x ≡ a ( mod q ) , $$\begin{align*} \hspace*{7pc}\sum _{\substack{n\leqslant x\\ n\equiv a\ (\mathrm{mod}\ q)}}f(n), \end{align*}$$ establishing that their variance over residue classes $a \ q)$ is small as soon = o $q=o(x)$ almost all moduli $q$ with nearly power-saving exceptional set . This improves and generalizes previous results Hooley on Barban–Davenport–Halberstam type theorems such moreover our essentially optimal unless one able to make progress certain well-known conjectures. are nevertheless prove stronger bounds number in cases where restricted be either smooth or prime, conditionally GRH we show estimate valid every These special “hybrid result” establish works short intervals arithmetic progressions simultaneously, thus generalizing Matomäki–Radziwiłł theorem intervals. also consider maximal deviation $a\ square root range 1 / 2 − ε $q\leqslant x^{1/2-\varepsilon }$ it “smooth-supported” again apart from providing smaller than what follows Bombieri–Vinogradov theorems. As an application methods, Linnik-type problems products exactly three primes, particular ternary approximation conjecture Erdős representing element group Z p × $\mathbb {Z}_p^{\times product two primes less $p$