Tatuzawa's theorem for Rankin-Selberg $L$-functions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909993841721344 |
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| author | Harcos, Gergely Thorner, Jesse |
| author_facet | Harcos, Gergely Thorner, Jesse |
| contents | Let $π$ and $π'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\varepsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\varepsilon)$ such that $L(s,π\timesπ')$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(π)C(π')(|\mathrm{Im}(s)|+1))^{\varepsilon}, \] where $C(π)$ and $C(π')$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,π\times\widetildeπ)$ by an idele class character $χ$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $π\otimesχ^2=π$. This extends earlier work of Humphries and Thorner. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10844 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tatuzawa's theorem for Rankin-Selberg $L$-functions Harcos, Gergely Thorner, Jesse Number Theory Primary 11M41, Secondary 11F66, 11F70 Let $π$ and $π'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\varepsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\varepsilon)$ such that $L(s,π\timesπ')$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(π)C(π')(|\mathrm{Im}(s)|+1))^{\varepsilon}, \] where $C(π)$ and $C(π')$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,π\times\widetildeπ)$ by an idele class character $χ$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $π\otimesχ^2=π$. This extends earlier work of Humphries and Thorner. |
| title | Tatuzawa's theorem for Rankin-Selberg $L$-functions |
| topic | Number Theory Primary 11M41, Secondary 11F66, 11F70 |
| url | https://arxiv.org/abs/2508.10844 |