Questions regarding the function $\Omega(n)$

Question

For any positive integer $n$, define $\Omega(n)$ to be the number of prime factors (including repeated factors, so for example $\Omega(12)=\Omega(2^2\times 3)=3$). It is well known (Pillai-Selberg) that $\Omega(n)$ distributes evenly across residues to any modulus. That is, for example with modulus $2$ we have that $\mathbb{P}(\Omega(n)\textrm{ is even})=1/2$. My question is - are there other well known results like this for $\Omega(n)$? For example, is the probability that $\Omega(n)$ is squarefree equal to the probability of an integer being squarefree?