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Given a partition $\{E_0,\ldots,E_n\}$ of the set of primes and a vector$\mathbf{k} \in \mathbb{N}_0^{n+1}$, we compute an asymptotic formula for thequantity $|\{m \leq x: \omega_{E_j}(m) = k_j \ \forall \ 0 \leq j \leq n\}|$uniformly in a wide range of the parameters $k_j$ that complements the resultsof a previous paper of the author. This is accomplished using an extension andgeneralization of a theorem of Wirsing due to the author that gives explicitestimates for the ratio $\frac{|M_g(x)|}{M_{f}(x)}$, whenever $f: \mathbb{N}\rightarrow (0,\infty)$ and $g: \mathbb{N} \rightarrow \mathbb{C}$ are stronglymultiplicative functions that are uniformly bounded on primes and satisfy$|g(n)| \leq f(n)$ for every $n \in \mathbb{N}$. This also allows us toconclude the validity of a probabilistic heuristic regarding$\pi(x;\mathbf{E},\mathbf{k})$ in the case that $k_j = (1+o(1))E_j(x)$, foreach $0 \leq j \leq n$.

On the Number of Restricted Prime Factors of an Integer II