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

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).