A Probabilistic Framework for the Erdos-Kac Theorem

Abstract

The Erdos-Kac theorem, a foundational result in probabilistic number theory, states that the number of prime factors of an integer follows a Gaussian distribution. In this paper we develop and analyze probabilistic models for "random integers" to study the mechanisms underlying this theorem. We begin with a simple model, where each prime p is chosen as a divisor with probability 1/p in a sequence of independent trials. A preliminary analysis shows that this construction almost surely yields an integer with infinitely many prime factors. To create a tractable framework, we study a truncated version Nx = product of p<=x of p^Xp, where Xp are independent Bernoulli(1/p) random variables. We prove an analogue of the Erdos-Kac theorem within this framework, showing that the number of prime factors omega(Nx) satisfies a central limit theorem with mean and variance asymptotic to log log x.