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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.07100 |
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| _version_ | 1866911257944129536 |
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| author | Garunkštis, Ramūnas Putrius, Jokūbas |
| author_facet | Garunkštis, Ramūnas Putrius, Jokūbas |
| contents | In this paper, we demonstrate the existence of the second moment of the Selberg zeta function for a Fuchsian group of the first kind at $σ= 1$. The prime geodesic theorem plays a crucial role in this context. The proof extends to Beurling zeta-functions satisfying a weak form of the Riemann hypothesis and to general Dirichlet series with positive coefficients, the partial sums of which are well-behaved. Note that by employing the recent approach of Broucke and Hilberdink in proving the second moment theorem, we can circumvent the separation condition introduced by Landau for general Dirichlet series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Selberg Zeta Functions Have Second Moment At $σ= 1$ Garunkštis, Ramūnas Putrius, Jokūbas Number Theory 11M36 In this paper, we demonstrate the existence of the second moment of the Selberg zeta function for a Fuchsian group of the first kind at $σ= 1$. The prime geodesic theorem plays a crucial role in this context. The proof extends to Beurling zeta-functions satisfying a weak form of the Riemann hypothesis and to general Dirichlet series with positive coefficients, the partial sums of which are well-behaved. Note that by employing the recent approach of Broucke and Hilberdink in proving the second moment theorem, we can circumvent the separation condition introduced by Landau for general Dirichlet series. |
| title | Selberg Zeta Functions Have Second Moment At $σ= 1$ |
| topic | Number Theory 11M36 |
| url | https://arxiv.org/abs/2511.07100 |