Average nonvanishing of Dirichlet $L$-functions at the central point

The generalized Riemann hypothesis implies that at least 50% of the central values L(12,χ) are nonvanishing as χ ranges over primitive characters modulo q. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo q and averages q over an interval, then at least 50.073% of the central values are nonvanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec.