2025/12/15 by Tony Haddad, Haddad, Tony
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2512.13669
openalex publication_date 2025/12/15 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28
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 ℓ1(\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 → ∞ as x → ∞.