A Refinement of the Burgess Bound for Character Sums

Abstract

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