On the Number of Restricted Prime Factors of an Integer II

Abstract

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 $|\{m\le x:{\omega }_{E_j}(m)=k_j\forall 0\le j\le n\}|$ 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 $f:\mathbb{N}\to (0,\infty )$ and $g:\mathbb{N}\to \mathbb{C}$ are strongly multiplicative functions that are uniformly bounded on primes and satisfy $|g(n)|\le f(n)$ 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 $0\le j\le n$.

14 pages; contents used to be part of preprint with ID arXiv:1604.00295, math.NT

References in corpus (2)

• On the Distribution of Integers with Restricted Prime Factors I
• A Strengthening of Theorems of Halász and Wirsing

Cited by in corpus (1)

• A Strengthening of Theorems of Halász and Wirsing