Non-vanishing and One Level Density for Dirichlet $L$-functions Along Short Averages
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918192507518976 |
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| author | Basak, Debmalya |
| author_facet | Basak, Debmalya |
| contents | Assuming the Generalized Riemann Hypothesis, it is known that at least half of the central values $L(\frac{1}{2},χ)$ are non-vanishing as $χ$ ranges over primitive characters modulo $q$. Unconditionally, this is known on average over both $χ$ modulo $q$ and $Q/2 \leq q \leq 2Q$. We prove that for any $δ>0$, there exist $η_1,η_2>0$ depending on $δ$ such that the non-vanishing proportion for $L(\frac{1}{2},χ)$ as $χ$ ranges modulo $q$ with $q$ varying in short intervals of size $Q^{1-η_1}$ around $Q$ and in arithmetic progressions with moduli up to $Q^{η_2}$ is larger than $\frac{1}{2}-δ$. Furthermore, by studying the one-level density of low-lying zeros of $L(s, χ)$, 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 $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12474 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-vanishing and One Level Density for Dirichlet $L$-functions Along Short Averages Basak, Debmalya Number Theory Assuming the Generalized Riemann Hypothesis, it is known that at least half of the central values $L(\frac{1}{2},χ)$ are non-vanishing as $χ$ ranges over primitive characters modulo $q$. Unconditionally, this is known on average over both $χ$ modulo $q$ and $Q/2 \leq q \leq 2Q$. We prove that for any $δ>0$, there exist $η_1,η_2>0$ depending on $δ$ such that the non-vanishing proportion for $L(\frac{1}{2},χ)$ as $χ$ ranges modulo $q$ with $q$ varying in short intervals of size $Q^{1-η_1}$ around $Q$ and in arithmetic progressions with moduli up to $Q^{η_2}$ is larger than $\frac{1}{2}-δ$. Furthermore, by studying the one-level density of low-lying zeros of $L(s, χ)$, 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 $q$. |
| title | Non-vanishing and One Level Density for Dirichlet $L$-functions Along Short Averages |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.12474 |