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arXiv · 2601.10927

Mixed incomplete character sums of rational functions with smooth moduli

Todd Cochrane·Andrew Granville·Junren Zheng
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Abstract

Let $χ=χ_q$ be a primitive character mod $q$ and fix $Δ>0$. In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M<n\leq M+N} χ(n)$ in intervals of length $N=q^Δ$ whenever $q$ is squarefree and is sufficiently smooth. Here we show that the smoothness parameter can be taken to be $N^{1-ε}$. We also discuss various generalizations and applications, obtaining best possible results in several aspects.

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Todd Cochrane, Andrew Granville, Junren Zheng. 2026-01-16. Mixed incomplete character sums of rational functions with smooth moduli. https://arxiv.org/abs/2601.10927

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