Burgess' bounds with prime moduli

Bryce Kerr; James Punch; Tim Trudgian

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arXiv · 2611.08816

Burgess' bounds with prime moduli

Bryce Kerr·James Punch·Tim Trudgian

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Abstract

Let $S_χ(N)= \sum_{n\leq N} χ(n)$, where $χ$ is a Dirichlet character modulo a prime $p$. For integral $r\geq 2$, Burgess gave bounds of the shape $S_χ(N) \ll N^{1-1/r} p^{(r+1)/(4r^{2})} \log p$. The factor of $\log p$ was reduced by several authors. We show that it can be removed completely.

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Bryce Kerr, James Punch, Tim Trudgian. 2026-11-06. Burgess' bounds with prime moduli. https://arxiv.org/abs/2611.08816

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