Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant $k$ facteurs premiers
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909866343268352 |
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| author | Garçonnet, Olivier |
| author_facet | Garçonnet, Olivier |
| contents | Let $x\geqslant 3$, for $1\leqslant n \leqslant x$ an integer, let $ω(n)$ be its number of distinct prime factors. We show that, among the values $n\leqslant x$ with $ω(n)=k$ where $1\leqslant k \ll \log_2 x$, $ω(n-1)$ satisfies an Erdős-Kac type theorem around $2\log_2 x$, so in large deviation regime, when weighted by $2^{ω(n-1)}$. This sharpens a result of Gorodetsky and Grimmelt with a quantitative and quasi-optimal error term. The proof of the main theorem is based on the characteristic function method and uses recent progress on Titchmarsh's divisor problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11616 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant $k$ facteurs premiers Garçonnet, Olivier Number Theory 11N25 (Primary), 11N05, 11N37, 11N13 Let $x\geqslant 3$, for $1\leqslant n \leqslant x$ an integer, let $ω(n)$ be its number of distinct prime factors. We show that, among the values $n\leqslant x$ with $ω(n)=k$ where $1\leqslant k \ll \log_2 x$, $ω(n-1)$ satisfies an Erdős-Kac type theorem around $2\log_2 x$, so in large deviation regime, when weighted by $2^{ω(n-1)}$. This sharpens a result of Gorodetsky and Grimmelt with a quantitative and quasi-optimal error term. The proof of the main theorem is based on the characteristic function method and uses recent progress on Titchmarsh's divisor problem. |
| title | Théorème d'Erdős-Kac dans un régime de grande déviation pour les translatés d'entiers ayant $k$ facteurs premiers |
| topic | Number Theory 11N25 (Primary), 11N05, 11N37, 11N13 |
| url | https://arxiv.org/abs/2410.11616 |