On square-free numbers generated from given sets of primes II

TL;DR

This paper investigates the asymptotic behavior of square-free numbers generated from specific prime sets, especially under conditions involving Dirichlet characters, extending previous research on their distribution.

Contribution

It advances the understanding of how square-free numbers generated from prime sets behave asymptotically when constrained by Dirichlet characters.

Findings

Asymptotic formulas for $Q_{\mathcal{P}}(x)$ under Dirichlet character conditions

Analysis of the distribution of square-free numbers in specified prime sets

Extension of previous results to more general prime set conditions

Abstract

We progress with the investigation started in article \cite{Roman2022}, namely the analysis of the asymptotic behaviour of $Q_{\mathcal{P}}(x)$ for different sets $\mathcal{P}$, 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 $\mathcal{P}$. We study how $Q_{\mathcal{P}}(x)$ behaves when we require that $\chi(p) = 1$ must hold for every $p \in \mathcal{P}$, where $\chi$ is a Dirichlet character.