Negative discrete second moments of Dirichlet $L$-functions

Abstract

Let $χ$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,χ)$ and that the non-trivial zeros $ρ=\tfrac12+iγ$ of $L(s,χ)$ are simple, we prove lower bounds for the discrete moments $\sum_{0<γ\le T}|L'(ρ,χ)|^{-2}$ and $\sum_{0<γ\le T}|L(2ρ,χ^2)/L'(ρ,χ)|^2$, uniformly in the conductor. The bounds capture the proportion $β/(1+β)$ of the conjectured asymptotics, where $β=\log T/\log qT$: 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 $q$.