An elementary Tauberian proof of the Prime Number Theorem
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909175871700992 |
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| author | Angot, Philippe |
| author_facet | Angot, Philippe |
| contents | We give a simple Tauberian proof of the Prime Number Theorem using only elementary real analysis. Hence, the analytic continuation of Riemann's zeta function $ζ$ and its non-vanishing value on the whole line $\{z\in {\mathbb C};\,{\mathrm{Re}\,} z=1\}$ are no more required. This is achieved by showing a strong extension for Laplace transforms on the real line of Wiener--Ikehara's theorem on Dirichlet's series, where the Tauberian assumption is reduced to a local boundary behavior around the pole. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An elementary Tauberian proof of the Prime Number Theorem Angot, Philippe Number Theory History and Overview 11A41, 11M45, 40A05, 40E05, 44A10 (primary), 11N05, 30B50, 42A38 (secondary) We give a simple Tauberian proof of the Prime Number Theorem using only elementary real analysis. Hence, the analytic continuation of Riemann's zeta function $ζ$ and its non-vanishing value on the whole line $\{z\in {\mathbb C};\,{\mathrm{Re}\,} z=1\}$ are no more required. This is achieved by showing a strong extension for Laplace transforms on the real line of Wiener--Ikehara's theorem on Dirichlet's series, where the Tauberian assumption is reduced to a local boundary behavior around the pole. |
| title | An elementary Tauberian proof of the Prime Number Theorem |
| topic | Number Theory History and Overview 11A41, 11M45, 40A05, 40E05, 44A10 (primary), 11N05, 30B50, 42A38 (secondary) |
| url | https://arxiv.org/abs/2404.13019 |