A new zero-free region for Rankin-Selberg $L$-functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913816631050240 |
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| author | Harcos, Gergely Thorner, Jesse |
| author_facet | Harcos, Gergely Thorner, Jesse |
| contents | Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Siegel's celebrated work on Dirichlet $L$-functions. As an application, we prove the first unconditional Siegel-Walfisz theorem for the Dirichlet coefficients of $-L'(s,π\timesπ')/L(s,π\timesπ')$. Also, for $n\leq 8$, we extend the region of holomorphy and nonvanishing for the twisted symmetric power $L$-functions $L(s,π,\mathrm{Sym}^n\otimesχ)$ of any cuspidal automorphic representation of $\mathrm{GL}(2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_16889 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new zero-free region for Rankin-Selberg $L$-functions Harcos, Gergely Thorner, Jesse Number Theory Primary 11M41, Secondary 11F66, 11F70 Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Siegel's celebrated work on Dirichlet $L$-functions. As an application, we prove the first unconditional Siegel-Walfisz theorem for the Dirichlet coefficients of $-L'(s,π\timesπ')/L(s,π\timesπ')$. Also, for $n\leq 8$, we extend the region of holomorphy and nonvanishing for the twisted symmetric power $L$-functions $L(s,π,\mathrm{Sym}^n\otimesχ)$ of any cuspidal automorphic representation of $\mathrm{GL}(2)$. |
| title | A new zero-free region for Rankin-Selberg $L$-functions |
| topic | Number Theory Primary 11M41, Secondary 11F66, 11F70 |
| url | https://arxiv.org/abs/2303.16889 |