Abstract: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 $x \to \infty$.
From: Tony Haddad [view email]
[v1]
Mon, 15 Dec 2025 18:56:05 UTC (18 KB)
[v2]
Wed, 8 Apr 2026 13:37:35 UTC (20 KB)
[v3]
Wed, 26 Aug 2026 16:20:44 UTC (18 KB)
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