Large values of shifted mixed character sums

Néo Tardy

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arXiv · 2605.13715

Large values of shifted mixed character sums

Néo Tardy

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Abstract

We consider sums of the form $$F_\chi(\alpha,\beta;\theta) := \sum_{\alpha p<n\le\beta p}\chi(n)e(n\theta),$$ where $\chi$ is a non-principal Dirichlet character modulo a prime number $p$. We prove that $$ \sqrt p \log \log p \ll \max_{0 \le \theta < 1}{\left|F_\chi(\alpha,\beta;\theta)\right|} \ll \sqrt{p}\log p, $$ generalizing an old result of Montgomery as well as a recent result of Iggidr in two aspects: we allow general non-principal characters $\chi$, and we consider incomplete mixed character sums.

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Néo Tardy. 2026-05-13. Large values of shifted mixed character sums. https://arxiv.org/abs/2605.13715

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