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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2507.14566 |
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| _version_ | 1866908456832729088 |
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| author | Qi, Zhi |
| author_facet | Qi, Zhi |
| contents | In this paper, we consider the non-vanishing problem for the family of special Hecke--Maass $L$-values $ L (1/2+it_f, f) $ with $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). We prove that 33% of $L (1/2+it_f, f)$ for $ t_f \leqslant T$ do not vanish as $T \rightarrow \infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 33% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on short intervals $|t_f-T| \leqslant T^μ$ for any $0 < μ< 1$. However, it is a curious case that the Riemann hypothesis does not yield better result for small $0 < μ\leqslant 1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points Qi, Zhi Number Theory In this paper, we consider the non-vanishing problem for the family of special Hecke--Maass $L$-values $ L (1/2+it_f, f) $ with $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). We prove that 33% of $L (1/2+it_f, f)$ for $ t_f \leqslant T$ do not vanish as $T \rightarrow \infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 33% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on short intervals $|t_f-T| \leqslant T^μ$ for any $0 < μ< 1$. However, it is a curious case that the Riemann hypothesis does not yield better result for small $0 < μ\leqslant 1/2$. |
| title | On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.14566 |