Expected Values of $L$-functions Away from the Central Point
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929701288673280 |
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| author | David, Chantal Meisner, Patrick |
| author_facet | David, Chantal Meisner, Patrick |
| contents | We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the transition at the point $s=\frac{1}{2}$ of the bound for the size of an $L$-function implied by the Lindelöf hypothesis. We show that at $s=\frac{1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_15120 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Expected Values of $L$-functions Away from the Central Point David, Chantal Meisner, Patrick Number Theory 11M06, 11M38, 11R16, 11R58 We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the transition at the point $s=\frac{1}{2}$ of the bound for the size of an $L$-function implied by the Lindelöf hypothesis. We show that at $s=\frac{1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function. |
| title | Expected Values of $L$-functions Away from the Central Point |
| topic | Number Theory 11M06, 11M38, 11R16, 11R58 |
| url | https://arxiv.org/abs/2305.15120 |