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

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 $s \to 1$.