Joint Poisson distribution of prime factors in sets

TL;DR

This paper proves that the joint distribution of prime factors in disjoint prime sets for a random integer approximates independent Poisson distributions, providing convergence rates and bounds, with applications to conditioned prime factor counts.

Contribution

It establishes the convergence of prime factor counts to independent Poisson variables and derives universal bounds, advancing understanding of prime factor distributions in number theory.

Findings

Joint distribution converges to independent Poissons

Provides explicit convergence rate using Kubilius model

Universal Poisson-type upper bound for prime factors

Abstract

Given disjoint subsets $T_1,\ldots,T_m$ 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 $T_1,\ldots,T_m$ 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 $T_1,\ldots,T_m$ are unrestricted, and apply this to the distribution of the number of prime factors from a set $T$ given that $n$ has $k$ total prime factors.