Joint Poisson distribution of prime factors in sets. (23rd July 2022) - FORD, KEVIN
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.