On natural densities of sets of some type integers

TL;DR

This paper proves a formula for the natural density of integers with prime exponents in specified ranges and generalizes this result, contributing to the understanding of the distribution of such integers.

Contribution

The paper establishes a formula for the natural density of integers with prime exponents in given intervals and extends this result to more general cases.

Findings

Derived the density formula for integers with prime exponents in specified ranges.

Proved the limit exists and equals the product over primes of certain sums.

Generalized the density formula to broader classes of sets.

Abstract

Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ 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: $$\lim_{x\to\infty}{\frac{Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)}{x}}=\prod\limits_{p}\sum\limits_{i=0}^{n}\left(\frac{1}{p^{a_{i}}}-\frac{1}{p^{b_{i}+1}}\right).$$ In this paper, we prove this result and then generalize it.