The Burgess inequality and the least k-th power non-residue

TL;DR

This paper provides a new explicit version of Burgess's inequality for character sums, applicable to any range, and improves bounds on least k-th power non-residues, with applications to prime size estimates.

Contribution

It introduces a universally applicable explicit estimate for the Burgess inequality and refines bounds on least k-th power non-residues for various character orders.

Findings

Explicit estimate for character sums valid for all M and N.

Improved bounds on least k-th power non-residues.

Applications to prime size thresholds for non-residues.

Abstract

The Burgess inequality is the best upper bound we have for the character sum $S_{\chi}(M,N) = \sum_{M<n\le M+N} \chi(n).$ Until recently, no explicit estimates had been given for the inequality. In 2006, Booker gave an explicit estimate for quadratic characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than $10^{140}$. Both of their explicit estimates are on restricted ranges. In this paper we prove an explicit estimate that works for any $M$ and $N$. We also improve McGown's estimates in a slightly narrower range, getting explicit estimates for characters of any order. We apply the estimates to the question of how large must a prime $p$ be to ensure that there is a $k$-th power non-residue less than $p^{1/6}$.