Abstract

Let χ be a Dirichlet character modulo p, a prime. In applications, one often needs estimates for short sums involving χ. 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 Treviño [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 pα for several fixed α. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields found in [7].

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