A refinement of the Burgess bound for character sums

TL;DR

This paper improves the Burgess bound for multiplicative character sums modulo a prime by refining the logarithmic factor, using a novel averaging method over numbers with no small prime factors.

Contribution

It introduces a new averaging technique over numbers without small prime factors to sharpen the Burgess bound for character sums.

Findings

Refined the Burgess bound with a smaller logarithmic factor.

Achieved sharper bounds for character sums for any nontrivial character.

Demonstrated the effectiveness of averaging over numbers with no small prime factors.

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 $\chi$ modulo a prime $q$ and any integer $r\ge 2$, we show that $$ \sum_{M<n\le M+N}\chi(n) = O\left( N^{1-1/r}q^{(r+1)/4r^2}(\log q)^{1/4r}\right), $$ 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.