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 integers, showing that under certain conditions, the number of distinct prime factors remains asymptotically normal.

Contribution

It generalizes the classical Erd ext{o}s-Kac Theorem to non-uniform distributions with perturbations, including Harmonic and Zipf distributions, establishing conditions for normal convergence.

Findings

Number of prime factors is asymptotically normal under generalized distributions.

Conditions on distribution perturbations ensure the Erd ext{o}s-Kac} behavior.

Results apply to Harmonic and Zipf distributions as special cases.

Abstract

Given $n\in\mathbb{N}$, 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 $$P_{n}\left(m\le n:\omega\left(m\right)-\log\log n\le x\left(\log\log n\right)^{1/2}\right)\to\mathbb{P}\left(Z\le x\right)$$ as $n\to\infty$; i.e., 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 $\frac{1}{n}$ in the following sense. Denote by $\mathbb{P}_{n}$ a probability distribution on $\left\{ 1,\ldots,n\right\} $ given by $\mathbb{P}_{n}\left(i\right)=\frac{1}{n}+\varepsilon_{i,n}$. By providing some constraints on the $\varepsilon_{i,n}$'s, sufficient conditions are stated in order to conclude that $$\mathbb{P}_{n}\left(m\le n:\omega\left(m\right)-\log\log n\le x\left(\log\log n\right)^{1/2}\right) \to \mathbb{P}\left(Z\le x\right)$$ as $n\to\infty.$ The main result will be applied to prove that the number of distinct prime factors of a positive integer with either the Harmonic$\left(n\right)$ distribution or the Zipf$\left(n,s\right)$ distribution also tends to the normal distribution $\mathcal{N}\left(\log\log n,\log\log n\right)$ as $n\to\infty$ (and as $s\to1$ in the case of a Zipf variable).