Correlations of multiplicative functions in function fields

TL;DR

This paper develops methods to analyze correlations of multiplicative functions over function fields, extending key conjectures and theorems from number theory to the setting of $\

Contribution

It introduces a new approach to study character sums with multiplicative functions over $\

Findings

Extended Matom"aki-Radziwill theorem to function fields.

Proved a version of Tao's two-point logarithmic Elliott conjecture.

Provided a new proof of K"atai's conjecture in the function field setting.

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 \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 a different phenomenon, specifically a low characteristic issue in the case that $q$ is a power of $2$. 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 use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a "corrected" form of the Erd\H{o}s discrepancy problem over $\mathbb{F}_q[t]$.