Negative discrete second moments of Dirichlet $L$-functions

Abstract

Let $\ddot{I}‡$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,\ddot{I}‡)$ and that the non-trivial zeros $\ddot{I}=\frac{1}{2}+i{\hat{I}}^3$ of $L(s,\ddot{I}‡)$ are simple, we prove lower bounds for the discrete moments ${\sum }_{0<{\hat{I}}^3\le T}|L^{\prime }(\ddot{I},\ddot{I}‡){|}^{-2}$ and ${\sum }_{0<{\hat{I}}^3\le T}|L(2\ddot{I},\ddot{I}{‡}^2)/L^{\prime }(\ddot{I},\ddot{I}‡){|}^2$, uniformly in the conductor. The bounds capture the proportion ${\hat{I}}^2/(1+{\hat{I}}^2)$ of the conjectured asymptotics, where ${\hat{I}}^2=\log T/\log qT$: 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 $q$.