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

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