Joint Poisson distribution of prime factors in sets. (23rd July 2022)

• Record Type: Journal Article
• Authors: FORD, KEVIN
• Is Part Of: Mathematical proceedings of the Cambridge Philosophical Society. Volume 173:Part 1(2022)
• Page Start: 189
• Page End: 200
• Publication Date: 2022-07-23
• Subjects: 11N25; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0305004121000499
• Languages: English
• ISSNs: 0305-0041

Abstract

Abstarct: Given disjoint subsets T 1, …, Tm of "not too large" primes up to x, we establish that for a random integer n drawn from [1, x ], the m -dimensional vector enumerating the number of prime factors of n from T 1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T 1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.