I am interested in finding an asymptotic for the following summatory function over $\Omega(n)$, which counts the total number of prime factors of $n$ honoring their multiplicity:
$$\sum_{k<n, \gcd(k,n)=1}{\Omega(k)}$$
Therefore, I want to find an asymptotic for the summatory function of $\Omega(k)$, where $k$ runs over the integers relatively prime to $n$ less than $n$.
In Hardy and Wright, the following asymptotic is computed:
$$\sum_{n\leq x}{\Omega(n)}=x\log\log x + B_2x + o(x)$$
Where $B_2\approx 1.0345061758$
However, I do not know how to proceed, or if there is some reference where the asymptotic is already computed. Any help will be welcomed.
Thanks in advance!