Estimates for character sums and Dirichlet $L$-functions to smooth moduli

Abstract

We use the $q$-analogue of van der Corput's method to estimate short character sums to smooth moduli. If $\chi$ is a primitive Dirichlet character modulo a squarefree, $q^\delta$-smooth integer $q$ we show that $$L(\frac12,\chi)\ll_\epsilon q^{\frac{27}{164}+O(\delta)+\epsilon}.$$