The natural density of some sets of square-free numbers

TL;DR

This paper derives a formula for the natural density of certain square-free numbers based on divisibility conditions and shows that this density is zero when primes are restricted by a congruence condition.

Contribution

It provides a simple formula for the density of square-free numbers with specified prime divisibility constraints, including the case of primes in a given residue class.

Findings

Derived a formula for the density of specific square-free numbers.

Showed the density is zero when primes are in a certain residue class.

Established conditions under which the density can be explicitly calculated.

Abstract

Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$ is the set of primes congruent to $r$ modulo $m$ (where $m$ and $r$ are relatively prime numbers), then this natural density is shown to be $0$.