Correlations of multiplicative functions in function fields

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen

Published 2020-09-28Version 1

We develop an approach to study character sums, weighted by a multiplicative function $f:\mathbb{F}_q[t]\to S^1$, of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)\chi(G)\xi(G), \end{equation} 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 $\mathbb{F}_q[t]$.