Explicit Burgess inequalities for cubefree moduli

TL;DR

This paper derives explicit Burgess inequalities for cubefree moduli, improving constants for r=2 and providing bounds for r≥3, with applications in number theory problems like power residues and L-functions.

Contribution

It improves explicit constants for Burgess inequalities with cubefree moduli, especially for r=2, and extends bounds to r≥3, enhancing tools for analytic number theory.

Findings

Improved explicit constant for r=2 case.

Derived explicit bounds for cubefree moduli with r≥3.

Applications to power residues and Dirichlet L-functions.

Abstract

Burgess proved that for $\chi_q$ a primitive Dirichlet character modulo $q$ with $q$ cubefree, $\Big|\sum_{M< n\le M+N}\chi_q(n)\Big| \ll N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+\epsilon}$ for all integers $r\ge1.$ More recently, explicit versions with prime moduli $q$ were computed by Booker, McGown, Trevi\~{n}o, and Francis, with applications to finding the least $k$-th power residue, and bounding the size of Dirichlet $L$-functions just to name a few. Jain-Sharma, Khale, and Liu proved an explicit estimate for $r=2.$ We improve their explicit constant for $r = 2$ and compute an explicit Burgess bound for cubefree $q$ for $r\ge 3$.