Short character sums for composite moduli

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square free moduli and also on the result due to Gallagher and Iwaniec when the core $q^{\mathrm{\prime }}={\prod }_{p\mathrm{\mid }q}p$ of the modulus $q$ satisfies $\log q^{\mathrm{\prime }}\sim \log q$. Some applications to zero free regions of Dirichlet L-functions and the $\mathrm{P}\acute{\mathrm{o}}\mathrm{l}\mathrm{y}\mathrm{a}$ and Vinogradov inequalities are indicated.