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

Given a natural number n, let ω(n) denote the number of distinct prime factors of n, let Z denote a standard normal variable, and let Pₙ denote the uniform distribution on \ 1,…,n\. The Erd{o}s-Kac Theorem states that if N(n) is a uniformly distributed variable on 1,…,n, then ω(N(n)) is asymptotically normally distributed as n→ ∞ with both mean and variance equal to log log n. The contribution of this paper is a generalization of the Erd{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 Pₙ a probability distribution on \ 1,…,n\ given by Pₙ(i)=1/n+εi,n. We provide sufficient conditions on εi,n so that the number of distinct prime factors of a Pₙ-distributed random variable is asymptotically normally distributed, as n→ ∞, 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(n) distribution also tends to the normal distribution, as n→ ∞. In addition, we explore sequences of distributions on the natural numbers such that ω(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(s) distributions as the parameter s → 1.