Negative discrete second moments of Dirichlet $L$-functions

Abstract

Let $\chi$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,\chi)$ and that the non-trivial zeros $\rho=\tfrac12+i\gamma$ of $L(s,\chi)$ are simple, we prove lower bounds for the discrete moments $\sum_{0<\gamma\le T}|L'(\rho,\chi)|^{-2}$ and $\sum_{0<\gamma\le T}|L(2\rho,\chi^2)/L'(\rho,\chi)|^2$, uniformly in the conductor. The bounds capture the proportion $\beta/(1+\beta)$ of the conjectured asymptotics, where $\beta=\log T/\log qT$: this is one half whenever $\log q=o(\log T)$, recovering for fixed $q$ the Dirichlet analogues of theorems of Milinovich and Ng and of Sinha, and degrades to $1/(2+A)$ when $q=T^{A}$. We conjecture the true leading order asymptotics and their analogues when we average over the family of primitive characters modulo $q$.