For any real number $x$, let's define $Om\_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, and $\Omega(n)$ the total number of prime factors of $n$ counted with multiplicity. Obviously $Om\_{0}(x)$ is just the number of squarefree integers below $x$.  

Do we know asymptotics and (maybe conditional) error terms for $Om\_{k}(x)$? What would be the consequences of RH on such error terms?  

Thanks in advance.



Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions
This problem was studied by Renyi, in


*On the density of certain sequences of integers*  

Publ. Inst. Math. Belgrade **8** (1955) 157-162.  

<http://elib.mi.sanu.ac.rs/files/journals/publ/14/13.pdf>


Let $d\_k = \lim\_{x\to\infty} Om\_k(x)/x$. Then the $d\_k$ are the coefficients in the following beautiful identity, valid for $|z| < 2$:
$$ \sum\_{k=0}^{\infty} d\_k z^k = \prod\_{p} \left(1-\frac{1}{p}\right)\left(1+\frac{1}{p-z}\right). $$


There have been several papers concerned with error estimates; a recent specimen is the following article by J. Wu:


*Sur un problème de Rényi*  

Monatsh. Math. **117** (1994), no. 3-4, 303–322. 


