Highly Uniform Prime Number Theorems
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866917957319262208 |
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| author | Kaneko, Ikuya Thorner, Jesse |
| author_facet | Kaneko, Ikuya Thorner, Jesse |
| contents | We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem depending explicitly on the available zero-free region. The class contains the Rankin-Selberg $L$-function $L(s,π\times π')$ associated to cuspidal automorphic representations $π$ and $π'$ of $\mathrm{GL}_{m}$ and $\mathrm{GL}_{m'}$, respectively. Our main result implies the first uniform prime number theorems for such $L$-functions (with analytic conductor uniformity) in complete generality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_09515 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Highly Uniform Prime Number Theorems Kaneko, Ikuya Thorner, Jesse Number Theory We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem depending explicitly on the available zero-free region. The class contains the Rankin-Selberg $L$-function $L(s,π\times π')$ associated to cuspidal automorphic representations $π$ and $π'$ of $\mathrm{GL}_{m}$ and $\mathrm{GL}_{m'}$, respectively. Our main result implies the first uniform prime number theorems for such $L$-functions (with analytic conductor uniformity) in complete generality. |
| title | Highly Uniform Prime Number Theorems |
| topic | Number Theory |
| url | https://arxiv.org/abs/2203.09515 |