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arXiv · 2411.14181

Average sizes of mixed character sums

Victor Y. Wang·Max Wenqiang Xu
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Abstract

We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} \chi(n) e(n\theta) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $\theta$ satisfying a weak Diophantine condition, where $\chi$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\le r$. In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $\theta$. Certain quadratic Diophantine equations play a key role in the present paper.

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Victor Y. Wang, Max Wenqiang Xu. 2024-11-21. Average sizes of mixed character sums. https://doi.org/10.1017/prm.2026.10123

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