An Investigation Into Several 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 \emph{Burgess' bound}. In this paper, we explore several minor adjustments one can make to the work of Enrique Trevi\~no on explicit versions of Burgess' bound. For an application, we investigate the problem of the existence of a $k$th 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.