On natural densities of sets of some type integers

Abstract

Let $a_0=b_0=0$ and $0<a_1\le b_1<a_2\le b_2<\dots \le b_n$ be integers. Let $Q\left(x;{\bigcup }_{j=1}^n[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in ${\bigcup }_{j=1}^n[a_j,b_j]$. The following formula holds: $$\underset{x\to \infty }{\lim }\frac{Q\left(x;{\bigcup }_{j=1}^n[a_j,b_j]\right)}{x}=\prod _p\sum _{i=0}^n\left(\frac{1}{p^{a_i}}-\frac{1}{p^{b_i+1}}\right).$$ In this paper, we prove this result and then generalize it.

References in corpus (2)

• On certain arithmetic functions involving exponential divisors
• On the asymptotic densities of certain subsets of ${\mathbf{N}}^{\mathbf{k}}$