2014/12/09 by Enrique Treviño, Treviño, Enrique · 1 citation
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1412.3062
arxiv created 2015/03/11 · arxiv updated 2015/03/13
The Burgess inequality is the best upper bound we have for the character sum Sχ(M,N) = ∑M<n≤ M+N χ(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 10140. 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 p1/6.