Poisson-Dirichlet approximation for counting integers with divisors in an interval
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915922802900992 |
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| author | Haddad, Tony |
| author_facet | Haddad, Tony |
| contents | We give a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space. By combining this with Arratia's coupling, recently refined by Koukoulopoulos and the author, we obtain a general strategy to reduce the problem of finding an asymptotic formula for the number of integers whose prime factorization lies in any given subset of $\ell^1(\mathbb R)$, to bounding two key probabilities measuring proximity to the boundary of the subset in question.
We apply this strategy to obtain an asymptotic formula for counting integers in $[1, x]$ that have a divisor in an interval $(y, z)$ in the regime $z/y \to \infty$ as $x \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poisson-Dirichlet approximation for counting integers with divisors in an interval Haddad, Tony Number Theory Probability We give a simple inequality that compares the laws of two random variables taking values in a convex subset of a normed vector space. By combining this with Arratia's coupling, recently refined by Koukoulopoulos and the author, we obtain a general strategy to reduce the problem of finding an asymptotic formula for the number of integers whose prime factorization lies in any given subset of $\ell^1(\mathbb R)$, to bounding two key probabilities measuring proximity to the boundary of the subset in question. We apply this strategy to obtain an asymptotic formula for counting integers in $[1, x]$ that have a divisor in an interval $(y, z)$ in the regime $z/y \to \infty$ as $x \to \infty$. |
| title | Poisson-Dirichlet approximation for counting integers with divisors in an interval |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2512.13669 |