An investigation into explicit versions of Burgess' bound

Let chi be a Dirichlet character modulo p, a prime. In applications, one often needs estimates for short sums involving chi. One such estimate is the family of bounds known as Burgess' bound. In this paper, we explore several adjustments one can make to the work of Trevino [11] in obtaining explicit versions of Burgess' bound. For an application, we investigate the problem of the existence of a kth power non-residue modulo p which is less than p(alpha) for several fixed alpha. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields found in [7]. (C) 2021 Elsevier Inc. All rights reserved.