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Let $\chi$ be a Dirichlet character modulo $p$, a prime. In applications, oneoften needs estimates for short sums involving $\chi$. One such estimate is thefamily of bounds known as \emph{Burgess' bound}. In this paper, we exploreseveral minor adjustments one can make to the work of Enrique Trevi\~no onexplicit versions of Burgess' bound. For an application, we investigate theproblem of the existence of a $k$th power non-residue modulo $p$ which is lessthan $p^\alpha$ for several fixed $\alpha$. We also provide a quick improvementto the conductor bounds for norm-Euclidean cyclic fields.

An Investigation Into Several Explicit Versions of Burgess' Bound