A Generalization of the Erd\H{o}s-Kac Theorem

TL;DR

This paper extends the Erd ext{o}s-Kac Theorem to a broader class of probability distributions on natural numbers, showing that the number of distinct prime factors remains asymptotically normal under certain perturbations.

Contribution

It provides sufficient conditions for generalized distributions on \\{1,...,n\\} to exhibit normality in the count of prime factors, broadening the theorem's applicability.

Findings

Number of prime factors is asymptotically normal under perturbations.

Harmonic distribution on natural numbers also yields normal prime factor count.

Generalizes results involving Zeta distributions as s approaches 1.

Abstract

Given a natural number $n$, let $\omega\left(n\right)$ denote the number of distinct prime factors of $n$, let $Z$ denote a standard normal variable, and let $P_{n}$ denote the uniform distribution on $\left\{ 1,\ldots,n\right\} $. The Erd\H{o}s-Kac Theorem states that if $N\left(n\right)$ is a uniformly distributed variable on $\lbrace 1,\ldots,n \rbrace$, then $\omega\left(N\left(n\right)\right)$ is asymptotically normally distributed as $n\to \infty$ with both mean and variance equal to $\log \log n$. The contribution of this paper is a generalization of the Erd\H{o}s-Kac Theorem to a larger class of random variables by considering perturbations of the uniform probability mass $1/n$ in the following sense. Denote by $\mathbb{P}_{n}$ a probability distribution on $\left\{ 1,\ldots,n\right\} $ given by $\mathbb{P}_{n}\left(i\right)=1/n+\varepsilon_{i,n}$. We provide sufficient conditions on $\varepsilon_{i,n}$ so that the number of distinct prime factors of a $\mathbb{P}_{n}$-distributed random variable is asymptotically normally distributed, as $n\to \infty$, with both mean and variance equal to $\log \log n$. Our main result is applied to prove that the number of distinct prime factors of a positive integer with the Harmonic$\left(n\right)$ distribution also tends to the normal distribution, as $n\to \infty$. In addition, we explore sequences of distributions on the natural numbers such that $\omega(n)$ is normally distributed in the limit. In addition, one of our theorems and its corollaries generalize a result from the literature involving the limit of $Zeta\left(s\right)$ distributions as the parameter $s \to 1$.