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 $$L(\frac{1}{2},\chi ){\ll }_{\epsilon }{q}^{\frac{27}{164}+O(\delta )+\epsilon }.$$

27 pages, including a 4 page appendix by é. Fouvry, E. Kowalski and P. Michel