A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application
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| Format: | Preprint |
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2025
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| _version_ | 1866912402506776576 |
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| author | Sun, Qingfeng Wang, Hui Yu, Yanxue |
| author_facet | Sun, Qingfeng Wang, Hui Yu, Yanxue |
| contents | Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $χ_r$. Let $q$ be an odd prime with $(q,r)=1$
and let $χ$ be a primitive Dirichlet character modulus $q$ with $χ\neqχ_r$. In this paper, we prove an unconditional Selberg-type zero-density estimate for the family of twisted $L$-functions $L(s, f \otimes χ)$ in the critical strip. As an application, we establish an asymptotic formula for the even moments of the argument function $S(t, f \otimes χ)=π^{-1}\arg L(1/2+ıt, f\otimesχ)$ and prove a central limit theorem for its distribution over $χ$ of modulus $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application Sun, Qingfeng Wang, Hui Yu, Yanxue Number Theory 11F12, 11F66 Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $χ_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $χ$ be a primitive Dirichlet character modulus $q$ with $χ\neqχ_r$. In this paper, we prove an unconditional Selberg-type zero-density estimate for the family of twisted $L$-functions $L(s, f \otimes χ)$ in the critical strip. As an application, we establish an asymptotic formula for the even moments of the argument function $S(t, f \otimes χ)=π^{-1}\arg L(1/2+ıt, f\otimesχ)$ and prove a central limit theorem for its distribution over $χ$ of modulus $q$. |
| title | A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application |
| topic | Number Theory 11F12, 11F66 |
| url | https://arxiv.org/abs/2505.23573 |