Explicit Burgess inequalities for cubefree moduli

Abstract

Burgess proved that for $\ddot{I}{‡}_q$ a primitive Dirichlet character modulo $q$ with $q$ cubefree, $|{\sum }_{M<n\le M+N}\ddot{I}{‡}_q(n)|\ll N^{1-\frac{1}{r}}{q}^{\frac{r+1}{4r^2}+\hat{I}\mu }$ for all integers $r\ge 1.$ More recently, explicit versions with prime moduli $q$ were computed by Booker, McGown, Treviño, 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 $r\ge 3$.