Note on a conjecture of Hildebrand regarding friable integers
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866915217997299712 |
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| author | de la Bretèche, Régis Tenenbaum, Gérald |
| author_facet | de la Bretèche, Régis Tenenbaum, Gérald |
| contents | Hildebrand proved that the smooth approximation for the number $Ψ(x,y)$ of $y$-friable integers not exceeding $x$ holds for $y>(\log x)^{2+\varepsilon}$ under the Riemann hypothesis and conjectured that it fails when $y\leqslant (\log x)^{2-\varepsilon}$. This conjecture has been recently confirmed by Gorodetsky by an intricate argument. We propose a short, straight-forward proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_15004 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Note on a conjecture of Hildebrand regarding friable integers de la Bretèche, Régis Tenenbaum, Gérald Number Theory 11N25 Hildebrand proved that the smooth approximation for the number $Ψ(x,y)$ of $y$-friable integers not exceeding $x$ holds for $y>(\log x)^{2+\varepsilon}$ under the Riemann hypothesis and conjectured that it fails when $y\leqslant (\log x)^{2-\varepsilon}$. This conjecture has been recently confirmed by Gorodetsky by an intricate argument. We propose a short, straight-forward proof. |
| title | Note on a conjecture of Hildebrand regarding friable integers |
| topic | Number Theory 11N25 |
| url | https://arxiv.org/abs/2211.15004 |