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

Time: Wed 2019-09-25 11.00

Lecturer: Oleksiy Klurman, MPIM Bonn

Location: F11, KTH math department

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