Computing Dirichlet character sums to a power-full modulus

The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This leads to a power-saving if the modulus is smooth enough. As an application, a fast, and potentially practical, method to compute Dirichlet L-functions with complexity exponent 1/3 for smooth enough moduli is derived.