On the constant in Burgess' bound for the number of consecutive residues or non-residues

Abstract

We give an explicit version of a result due to D. Burgess. Let $χ$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $χ$ takes on a particular value is less than $\left\{\frac{πe\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ term is given explicitly.