On Arratia’s coupling and the Dirichlet law for the factors of a random integer
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| Titel: | On Arratia’s coupling and the Dirichlet law for the factors of a random integer |
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| Autoren: | Haddad, Tony, Koukoulopoulos, Dimitris |
| Quelle: | Journal de l’École polytechnique — Mathématiques. 12:1565-1604 |
| Publication Status: | Preprint |
| Verlagsinformationen: | Cellule MathDoc/Centre Mersenne, 2025. |
| Publikationsjahr: | 2025 |
| Schlagwörter: | 11N25, 11N37, 11N60, 60B12, Number Theory, Probability (math.PR), FOS: Mathematics, Number Theory (math.NT), Probability |
| Beschreibung: | Let x≥2, let N x be an integer chosen uniformly at random from the set ℤ∩[1,x], and let (V 1 ,V 2 ,...) be a Poisson–Dirichlet process of parameter 1. We prove that there exists a coupling of these two random objects such that𝔼∑ i≥1 |logP i -V i logx|≍1,where the implied constants are absolute and N x =P 1 P 2 ⋯ is the unique factorization of N x into primes or ones with the P i ’s being non-increasing. This establishes a 2002 conjecture of Arratia, who constructed a coupling for which the left-hand side in the above estimate is ≪loglogx, and who also proved that the left-hand side is ≥1-o(1) for all couplings. In addition, we use our refined coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into k parts proved in 2023 by Leung and we improve on its error term. |
| Publikationsart: | Article |
| Sprache: | English |
| ISSN: | 2270-518X |
| DOI: | 10.5802/jep.317 |
| DOI: | 10.48550/arxiv.2406.09360 |
| Zugangs-URL: | http://arxiv.org/abs/2406.09360 |
| Rights: | CC BY arXiv Non-Exclusive Distribution |
| Dokumentencode: | edsair.doi.dedup.....cf83fd01150aa6535821df6ca34a0c8e |
| Datenbank: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Let x≥2, let N x be an integer chosen uniformly at random from the set ℤ∩[1,x], and let (V 1 ,V 2 ,...) be a Poisson–Dirichlet process of parameter 1. We prove that there exists a coupling of these two random objects such that𝔼∑ i≥1 |logP i -V i logx|≍1,where the implied constants are absolute and N x =P 1 P 2 ⋯ is the unique factorization of N x into primes or ones with the P i ’s being non-increasing. This establishes a 2002 conjecture of Arratia, who constructed a coupling for which the left-hand side in the above estimate is ≪loglogx, and who also proved that the left-hand side is ≥1-o(1) for all couplings. In addition, we use our refined coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into k parts proved in 2023 by Leung and we improve on its error term. |
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| ISSN: | 2270518X |
| DOI: | 10.5802/jep.317 |