Correlations of multiplicative functions in function fields

Abstract

We develop an approach to study character sums, weighted by a multiplicative function f:Fq[t]S1f:\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 χ\chi is a Dirichlet character and ξ\xi is a short interval character over Fq[t].\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 Fq[t]\mathbb{F}_q[t], where qq 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 qq. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that qq is a power of 22. 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 Fq[t]\mathbb{F}_q[t].Comment: 62 pages; further referee comments incorporated; to appear in Mathematik

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