Central non-vanishing of Dirichlet $L$-functions

Abstract

We prove that, for every sufficiently large modulus $q\not\equiv 2\mathrm{mod}4$, at least $3/8-o(1)$ of the primitive Dirichlet characters $\chi$ modulo $q$ have $L(1/2,\chi )\ne 0$. We also establish stronger proportions for moduli $q$ satisfying certain arithmetic conditions.