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 $L(\frac{1}{2},\ddot{I}‡)$ are non-vanishing as $\ddot{I}‡$ ranges over primitive characters modulo $q$. Unconditionally, this is known on average over both $\ddot{I}‡$ modulo $q$ and $Q/2\le q\le 2Q$. We prove that for any $\hat{I}´>0$, there exist $\hat{I}{\cdot }_1,\hat{I}{\cdot }_2>0$ depending on $\hat{I}´$ such that the non-vanishing proportion for $L(\frac{1}{2},\ddot{I}‡)$ as $\ddot{I}‡$ ranges modulo $q$ with $q$ varying in short intervals of size $Q^{1-\hat{I}{\cdot }_1}$ around $Q$ and in arithmetic progressions with moduli up to $Q^{\hat{I}{\cdot }_2}$ is larger than $\frac{1}{2}-\hat{I}´$. Furthermore, by studying the one-level density of low-lying zeros of $L(s,\ddot{I}‡)$, 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 $q$.

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