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arXiv · 1711.10582

A refinement of the Burgess bound for character sums

Bryce Kerr·Igor E. Shparlinski·Kam Hung Yau
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Abstract

In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski. In particular, for any nontrivial multiplicative character $χ$ modulo a prime $q$ and any integer $r\ge 2$, we show that $$ \sum_{M<n\le M+N}χ(n) = O\left( N^{1-1/r}q^{(r+1)/4r^2}(\log q)^{1/4r}\right), $$ which sharpens previous results by a factor $(\log q)^{1/4r}$. Our improvement comes from averaging over numbers with no small prime factors rather than over an interval as in previous approaches.

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Bryce Kerr, Igor E. Shparlinski, Kam Hung Yau. 2019-05-07. A refinement of the Burgess bound for character sums. https://arxiv.org/abs/1711.10582

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