An Investigation Into Several Explicit Burgess Inequalities

Forrest J. Francis

Published 2019-10-30Version 1

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' inequality. In this paper, we explore several small adjustments one can make to the work of Enrique Trevi\~no on explicit versions of Burgess' inequality. These improvements can be used to show that, for $p > 10^{5}$, the least $k$th power non-residue modulo $p$ is no larger than $p^\frac{1}{6}$. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields.