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

TL;DR

This paper provides an explicit bound on the maximum length of consecutive integers where a non-principal Dirichlet character modulo a prime takes a constant value, refining Burgess's classical result with explicit constants.

Contribution

It offers an explicit version of Burgess's bound, including precise constants and an explicit error term, improving the understanding of character sum bounds.

Findings

Bound on consecutive residues is less than (πe√6/3 + o(1)) p^{1/4} log p

Explicit constants and error term provided for the bound

Refinement of Burgess's original asymptotic result

Abstract

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