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

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.