The Burgess inequality and the least kth power non-residue

Abstract

The Burgess inequality is the best upper bound we have for incomplete character sums of Dirichlet characters. In 2006, Booker gave an explicit estimate for quadratic Dirichlet 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 range. 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 kth power non-residue less than p1/6.

The Euclidean algorithm in quintic and septic cyclic fields

“…Finally, the remainder of the paper is devoted to supplying the necessary justification for the upper bound on the conductor given in Theorem 1.6. The proof involves applying some recent results of Treviño concerning non-residues (see [18,19,17]) together with ideas in [14,12]. For completeness, we provide improved (unconditional) conductor bounds for degrees ℓ > 3 as well.…”

“…3 In the technical condition appearing in the proposition, one must replace 4f 1/2 by 2f 1/2 ; however, in our application, this condition will be automatically met so this has no real effect. Moreover, it is shown in [17] that the technical condition may be dropped completely provided k ≥ 3.…”

Explicit bounds for small prime nonresidues

“…When k = 1, there is the work of Norton (see [8]) that was later superceded by Treviño (see [13]) and when k = 2 there is a paper by McGown (see [6]). The proof of our result involves a modification of McGown's work (see [6]), which is based on the method of Burgess (see [3,4]), and the adoption of Treviño's results (see [12]). Proof.…”

Explicit upper bounds on the least primitive root

“…for p sufficiently large. For example, when p ≥ 10 15 , Treviño [20] obtains the following constants in (1) which are the best known. We prove the following explicit upper bound for g(p) that not only improves these constants, but also removes the log term completely.…”