Zeros of Polynomials in Derivatives of Automorphic $L$-functions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912791096459264 |
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| author | Dong, Anji Wattanawanichkul, Nawapan Zaharescu, Alexandru |
| author_facet | Dong, Anji Wattanawanichkul, Nawapan Zaharescu, Alexandru |
| contents | Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbolπ)$ be a polynomial in the derivatives of $L$-functions associated with representations $π_u \in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbolπ)$ with $0 < \operatorname{Im}(s) < T$. We explicitly determine the main term of this formula in terms of the degrees, the ranks, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\boldsymbolπ)$ lie near the critical line $\operatorname{Re}(s)=1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of Polynomials in Derivatives of Automorphic $L$-functions Dong, Anji Wattanawanichkul, Nawapan Zaharescu, Alexandru Number Theory 11F66, 11M26 Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbolπ)$ be a polynomial in the derivatives of $L$-functions associated with representations $π_u \in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbolπ)$ with $0 < \operatorname{Im}(s) < T$. We explicitly determine the main term of this formula in terms of the degrees, the ranks, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\boldsymbolπ)$ lie near the critical line $\operatorname{Re}(s)=1/2$. |
| title | Zeros of Polynomials in Derivatives of Automorphic $L$-functions |
| topic | Number Theory 11F66, 11M26 |
| url | https://arxiv.org/abs/2512.22451 |