On the Number of Restricted Prime Factors of an Integer II

Abstract

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 the quantity $|\{m \leq x: \omega_{E_j}(m) = k_j \ \forall \ 0 \leq j \leq n\}|$ uniformly in a wide range of the parameters $k_j$ that complements the 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 $\frac{|M_g(x)|}{M_{f}(x)}$, 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 $|g(n)| \leq f(n)$ 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 $0 \leq j \leq n$.