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Oleksiy Klurman: Multiplicative functions in short arithmetic progressions and applications

Time: Wed 2019-09-25 11.00

Location: F11, KTH math department

Participating: Oleksiy Klurman, MPIM Bonn

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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).