On the Number of Restricted Prime Factors of an Integer II

Given a partition $\{E_0,\dots ,E_n\}$ of the set of primes and a vector $\mathbf{k}\in {\mathbb{N}}_0^{n+1}$, we compute an asymptotic formula for the quantity $\mathrm{\mid }\{m\le x:{\omega }_{E_j}(m)=k_j\mathrm{\forall }0\le j\le n\}\mathrm{\mid }$ uniformly in a wide range of the parameters $k_j$ that complements the results of a previous paper of the author. This is accomplished using an extension and generalization of a theorem of Wirsing due to the author that gives explicit estimates for the ratio $\frac{\mathrm{\mid }{M}_g(x)\mathrm{\mid }}{M_f(x)}$, whenever $f: $\mathbb{N}$ \rightarrow (0,\infty)$and$g: $\mathbb{N}$ \rightarrow $\mathbb{C}$$ are strongly multiplicative functions that are uniformly bounded on primes and satisfy $\mathrm{\mid }g(n)\mathrm{\mid }\le f(n)$ for every $n\in \mathbb{N}$. This also allows us to conclude the validity of a probabilistic heuristic regarding $\pi (x;\mathbf{E},\mathbf{k})$ in the case that $k_j=(1+o(1))E_j(x)$, for each $0\le j\le n$.