On the Hardy-Ramanujan Theorem

Abstract

In this note we prove an effective version of the Hardy--Ramanujan Theorem. For every $x\ge 2$ and every non-negative function $F$ on the non-negative integers, we show $$\frac{1}{x}\sum_{2\le n\le x}F(ω(n)-1)\le 118\,\mathbb{E}F(Z_{\log\log x+4.096}),$$ where $Z_λ$ is Poisson with parameter $λ$. Thus the shifted empirical distribution of $ω(n)$ is pointwise dominated by a fixed multiple of a Poisson law. We also obtain the sharper squarefree analogue, derive explicit Chernoff and Gaussian-window estimates, obtain moderate-deviation upper bounds and uniform moment estimates, and transfer these consequences to $Ω(n)$ and to the number of prime divisors occurring exactly once.