On square-free numbers generated from given sets of primes

TL;DR

This paper investigates the distribution of square-free numbers generated from specific prime sets within a given range, providing insights into their counting function and asymptotic behavior.

Contribution

It introduces a new analysis of the counting function for square-free numbers formed from prime sets constrained by a monotone function, extending understanding of their distribution.

Findings

Derived asymptotic estimates for $Q_{\ ext{\mathcal{P}}}(x)$

Established bounds for the count of such square-free numbers

Analyzed the influence of the prime set's size on distribution

Abstract

Let $x$ be a positive real number, and $\mathcal{P} \subset [2,\lambda(x)]$ be a set of primes, where $\lambda(x) \in \Omega(x^\varepsilon)$ is a monotone increasing function with $\varepsilon \in (0,1)$. We examine $Q_{\mathcal{P}}(x)$, where $Q_{\mathcal{P}}(x)$ is the element count of the set containing those positive square-free integers, which are smaller than-, or equal to $x$, and which are only divisible by the elements of $\mathcal{P}$.