Author
Klurman, O
Mangerel, A
Teräväinen, J
Last updated
2020-12-20T04:38:51.76+00:00
Abstract
We study for bounded multiplicative functions $f$ sums of the form
\begin{align*} \sum_{\substack{n\leq x \atop n\equiv a\pmod q}}f(n),
\end{align*} 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 substantially improves
on previous 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. We are nevertheless
able to prove stronger bounds for the number of the exceptional moduli $q$ in
the cases where $q$ is restricted to be either smooth or prime, and
conditionally on GRH we show that our variance estimate is valid for every $q$.
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 Matom\"aki-Radziwill theorem
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 Bombieri-Vinogradov-type theorems. As an application of our methods, we
consider the analogue of 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 $q$.
Symplectic ID
1061279
Download URL
http://arxiv.org/abs/1909.12280v3
Publication type
Journal Article
Please contact us with feedback and comments about this page. Created on 08 Oct 2019 - 17:30.