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 χ modulo a prime q and any integer r⩾2, we show that ∑M<n⩽M+Nχ(n)=O(N1−1/rq(r+1)/4r2(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.