Refinements to the prime number theorem for arithmetic progressions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | |
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| _version_ | 1866916161576239104 |
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| author | Thorner, Jesse Zaman, Asif |
| author_facet | Thorner, Jesse Zaman, Asif |
| contents | We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length $x^{1-δ}$, a Brun-Titchmarsh bound, and Linnik's bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov-Korobov zero-free region, a log-free zero density estimate, and the Deuring-Heilbronn zero repulsion phenomenon. Improvements exist when the modulus is sufficiently powerful. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_10878 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Refinements to the prime number theorem for arithmetic progressions Thorner, Jesse Zaman, Asif Number Theory 11N05 (Primary), 11N13, 11M06 (Secondary) We prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length $x^{1-δ}$, a Brun-Titchmarsh bound, and Linnik's bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov-Korobov zero-free region, a log-free zero density estimate, and the Deuring-Heilbronn zero repulsion phenomenon. Improvements exist when the modulus is sufficiently powerful. |
| title | Refinements to the prime number theorem for arithmetic progressions |
| topic | Number Theory 11N05 (Primary), 11N13, 11M06 (Secondary) |
| url | https://arxiv.org/abs/2108.10878 |