Correlations of multiplicative functions in function fields

Klurman, O
Mangerel, A
Teräväinen, J

Last Updated:

2020-10-25T09:09:53.107+00:00

abstract:

We develop an approach to study character sums, weighted by a multiplicative
function $f:\mathbb{F}_q[t]\to S^1$, of the form \sum_{G\in
\mathcal{M}_N}f(G)\chi(G)\xi(G), where $\chi$ is a Dirichlet
character and $\xi$ is a short interval character over $\mathbb{F}_q[t].$ We
then deduce versions of the Matom\"aki-Radziwill theorem and Tao's two-point
logarithmic Elliott conjecture over function fields $\mathbb{F}_q[t]$, where
$q$ is fixed. The former of these improves on work of Gorodetsky, and the
latter extends the work of Sawin-Shusterman on correlations of the M\"{o}bius
function for various values of $q$.
Compared with the integer setting, we encounter some different phenomena,
specifically a low characteristic issue in the case that $q$ is a power of $2$,
as well as the need for a wider class of ''pretentious" functions called Hayes
characters.
As an application of our results, we give a short proof of the function field
version of a conjecture of K\'atai on classifying multiplicative functions with
small increments, with the classification obtained and the proof being
different from the integer case.
In a companion paper, we will use these results to characterize the limiting
behavior of partial sums of multiplicative functions in function fields and in
particular to solve the ''corrected" form of the Erd\H{o}s discrepancy problem
over $\mathbb{F}_q[t]$.

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