Am I correct in assuming that the same result: $$ N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid \ \sim \frac{x}{\log x}\frac{(\log_2 x)^{k-1}}{(k-1)!}\ (x \rightarrow \infty) $$ also holds for: $$ \pi_k(x):=\ \mid\{n\leq x : \omega(n)=k\}\mid \ $$ even though they are wildly different for high $k$? See here.