On an unconditional $\rm GL_3$ analog of Selberg's result
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909929446572032 |
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| author | Sun, Qingfeng Wang, Hui |
| author_facet | Sun, Qingfeng Wang, Hui |
| contents | Let $F$ be a Hecke--Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ with the Langlands parameter $μ_{F}=\big(μ_{F,1},μ_{F,2},μ_{F,3}\big)$ and the associated $L$-function $L(s, F)$. Define $S_F(t)=π^{-1}\arg L(1/2+\mathrm{i}t, F)$. When $μ_{F}$ is in generic position, we establish an unconditional asymptotic formula for the moments of $S_F(t)$. Previously, such a formula was only known to hold under the Generalized Riemann Hypothesis. The key ingredient is a weighted zero-density estimate in the spectral aspect for $L(s, F)$, which has recently been proved by Sun and Wang in arXiv:2412.02416. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On an unconditional $\rm GL_3$ analog of Selberg's result Sun, Qingfeng Wang, Hui Number Theory 11F12, 11F66, 11F72 Let $F$ be a Hecke--Maass cusp form for $\mathrm{SL}_3(\mathbb{Z})$ with the Langlands parameter $μ_{F}=\big(μ_{F,1},μ_{F,2},μ_{F,3}\big)$ and the associated $L$-function $L(s, F)$. Define $S_F(t)=π^{-1}\arg L(1/2+\mathrm{i}t, F)$. When $μ_{F}$ is in generic position, we establish an unconditional asymptotic formula for the moments of $S_F(t)$. Previously, such a formula was only known to hold under the Generalized Riemann Hypothesis. The key ingredient is a weighted zero-density estimate in the spectral aspect for $L(s, F)$, which has recently been proved by Sun and Wang in arXiv:2412.02416. |
| title | On an unconditional $\rm GL_3$ analog of Selberg's result |
| topic | Number Theory 11F12, 11F66, 11F72 |
| url | https://arxiv.org/abs/2502.01288 |