# Oleksiy Klurman: Multiplicative functions in short arithmetic progressions and applications

**Time: **
Wed 2019-09-25 11.00

**Location: **
F11, KTH math department

**Lecturer: **
Oleksiy Klurman, MPIM Bonn

We study for bounded multiplicative functions f sums of the form

\sum_{\substack{n\leq x\\n\equiv a\pmod q}}f(n)

establishing a theorem stating 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 sharply improves

on previous classical 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.

These results are special cases of a ''hybrid result'' that we

establish that works for sums of f(n) over almost all short intervals

and arithmetic progressions simultaneously, thus generalizing the

breakthrough work of Matom\"aki and Radziwi\l{}\l{} on multiplicative

functions in short intervals.

We also consider the maximal deviation of f(n) over all residue

classes a\pmod q in the square-root range 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 the celebrated

Bombieri--Vinogradov-type theorems.

As an application of our methods, we consider the analogue of the

classical Linnik's theorem on the least prime in an arithmetic

progression for products of exactly three primes, and prove the

exponent 2+o(1)for this problem for all smooth values of q and for all

but a few prime moduli q. This is consistent with what was previously

known assuming GRH. This is based on a joint work with A. Mangerel

(CRM) and J. Teravainen (Oxford).