Hardy-Ramanujan Theorem

Let $\omega(n)$ be the number of Distinct Prime Factors of $n$. If $\Psi(x)$ tends steadily to infinity with $x$, then

$$\ln\ln x-\Psi(x)\sqrt{\ln\ln x}<\omega(n)<\ln\ln x+\Psi(x)\sqrt{\ln\ln x}$$

for Almost All numbers $n<x$. ``Almost All'' means here the frequency of those Integers $n$ in the interval $1\leq n\leq x$ for which

$$\vert\omega(n)-\ln\ln x\vert>\Psi(x)\sqrt{\ln\ln x}$$

approaches 0 as $x\to\infty$.

See also Distinct Prime Factors, Erdös-Kac Theorem