An Investigation Into Several Explicit Versions of Burgess' Bound

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 minor adjustments one can make to the work of Enrique Treviño [11] on 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 α for several fixed α. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields found in [7].

Explicit Improvements to the Burgess Bound Via Pólya-Vinogradov

“…Conversely, when working with explicit versions of these estimates, any improvement to the leading constant in the P-V inequality will immediately yield improvements in the leading constant for Burgess' bound (see, for example, [21] and [7]). Fromm and Goldmakher [9] have recently established that, in fact, improvements to the P-V inequality can be used to extract improvements to the effective range (with respect to t) in Burgess' bound.…”

Explicit Burgess Bound for Composite Moduli

“…In the prime modulus case, explicit Burgess-strength bounds have been worked out by Booker [4], McGown [15], Treviño [27], and Francis [11]. In particular, [27, Corollary 1], is a totally explicit version of the Burgess inequality as stated in (1) for all positive integers r and all prime moduli p ≥ 10 7 .…”