Non-vanishing and One Level Density for Dirichlet $L$-functions Along Short Averages

Abstract

Assuming the Generalized Riemann Hypothesis, it is known that at least half of the central values $L(\frac{1}{2},\chi)$ are non-vanishing as $\chi$ ranges over primitive characters modulo $q$. Unconditionally, this is known on average over both $\chi$ modulo $q$ and $Q/2 \leq q \leq 2Q$. We prove that for any $\delta>0$, there exist $\eta_1,\eta_2>0$ depending on $\delta$ such that the non-vanishing proportion for $L(\frac{1}{2},\chi)$ as $\chi$ ranges modulo $q$ with $q$ varying in short intervals of size $Q^{1-\eta_1}$ around $Q$ and in arithmetic progressions with moduli up to $Q^{\eta_2}$ is larger than $\frac{1}{2}-\delta$. Furthermore, by studying the one-level density of low-lying zeros of $L(s, \chi)$, we show that under the Generalized Riemann Hypothesis, non-vanishing proportions exceeding $\frac{1}{2}$ can be obtained while still averaging over short ranges of $q$.