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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2506.08546 |
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| _version_ | 1866908456572682240 |
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| author | Qi, Zhi |
| author_facet | Qi, Zhi |
| contents | Let $Q (z)$ be a holomorphic Hecke cusp newform of square-free level and $u_j (z)$ traverse an orthonormal basis of Hecke--Maass cusp forms of full level. Let $1/4 + t_j^2$ be the Laplace eigenvalue $u_j (z)$. In this paper, we prove that there is a constant $ γ(Q) $ expressed as a certain Euler product associated to $Q$ such that at least $ γ(Q) / 11 $ of the Rankin--Selberg special $L$-values $L (1/2+it_j, Q \otimes u_j)$ for $ t_j \leqslant T$ do not vanish as $T \rightarrow \infty$. Further, we show that the non-vanishing proportion is at least $γ(Q) \cdot (4μ-3) / (4μ+7) $ on the short interval $ |t_j - T| \leqslant T^μ $ for any $3/4 < μ< 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08546 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points Qi, Zhi Number Theory Let $Q (z)$ be a holomorphic Hecke cusp newform of square-free level and $u_j (z)$ traverse an orthonormal basis of Hecke--Maass cusp forms of full level. Let $1/4 + t_j^2$ be the Laplace eigenvalue $u_j (z)$. In this paper, we prove that there is a constant $ γ(Q) $ expressed as a certain Euler product associated to $Q$ such that at least $ γ(Q) / 11 $ of the Rankin--Selberg special $L$-values $L (1/2+it_j, Q \otimes u_j)$ for $ t_j \leqslant T$ do not vanish as $T \rightarrow \infty$. Further, we show that the non-vanishing proportion is at least $γ(Q) \cdot (4μ-3) / (4μ+7) $ on the short interval $ |t_j - T| \leqslant T^μ $ for any $3/4 < μ< 1$. |
| title | On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.08546 |