On the Hardy-Ramanujan Theorem

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 $$\frac{1}{x}\sum _{2\le n\le x}F(\omega (n)-1)\le 118\mathbb{E}F(Z_{\log \log x+4.096}),$$ where $Z_{\lambda }$ is Poisson with parameter $\lambda$. Thus the shifted empirical distribution of $\omega (n)$ 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 $\mathrm{\Omega }(n)$ and to the number of prime divisors occurring exactly once.