Spacings between integers having typically many prime factors

TL;DR

This paper demonstrates that integers with a typical number of prime factors are distributed in a way that their spacings follow a Poisson process, revealing a probabilistic structure in their distribution.

Contribution

It establishes that integers with near-typical prime factor counts form a Poisson process, extending understanding of prime factor distribution.

Findings

Nearest neighbor spacings follow Poisson distribution

Integers with typical prime factors exhibit Poisson process behavior

Results hold for arbitrarily small positive bb

Abstract

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\de$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $|\om(n)-\log_2 n|\le (\log_2 n)^{\de}$ obey the Poisson distribution law.