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Correlations of multiplicative functions in function fields
- Source :
- Mathematika, 69 (2023), 155-231
- Publication Year :
- 2020
-
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 $\mathbb{F}_q[t]$.<br />Comment: 62 pages; further referee comments incorporated; to appear in Mathematika
- Subjects :
- Mathematics - Number Theory
11N37, 11T55
Subjects
Details
- Database :
- arXiv
- Journal :
- Mathematika, 69 (2023), 155-231
- Publication Type :
- Report
- Accession number :
- edsarx.2009.13497
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1112/mtk.12181