Correlations of multiplicative functions in function fields

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)χ(G)ξ(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 Matomäki-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 Möbius 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 Kátai 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 Erdős discrepancy problem over ${\mathbb{F}}_q[t]$.

arXiv comment62 pages; further referee comments incorporated; to appear in Mathematika

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Cited by in corpus (2)

• Multiplicative functions in short arithmetic progressions
• Beyond the Erdős discrepancy problem in function fields