Explicit Improvements to the Burgess Bound Via P\'olya-Vinogradov

TL;DR

This paper improves the Burgess bound for short character sums by explicitly refining constants in the Pólya-Vinogradov inequality, supported by explicit estimates of multiplicative functions and the Dickman function.

Contribution

It provides explicit versions of key estimates that enable better constants in the Burgess bound through the Pólya-Vinogradov inequality.

Findings

Explicit bounds for mean values of real multiplicative functions

Refined estimates of the Dickman function

Improved constants in the Burgess bound

Abstract

We make explicit a theorem of Fromm and Goldmakher [arXiv:1706.03002], which states that one can improve Burgess' bound for short character sums simply by improving the leading constant in the P\'{o}lya-Vinogradov inequality. Towards achieving this, we establish explicit versions of several estimates related to the mean values of real multiplicative functions and the Dickman function.