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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 $$L\left({1 \over 2},\chi\right)\ll_\epsilon q^{{27 \over 164}+O\left(\delta\right)+\epsilon}.$$

Estimates for Character Sums and DirichletL-Functions to Smooth Moduli