Numbers without prime factors in a set of positive relative density

Question

Let $\mathcal{P}$ be a set of prime numbers of relative density $\kappa \in (0, 1)$, which means that $$\#\left(\mathcal{P} \cap [1, x]\right) = \kappa \,\pi(x) + E(x) \quad (x \to \infty)$$ for a "suitable" error term $E(x)$ (of course, $E(x) = o(\pi(x))$).

Let $\mathcal{S}$ be the set of natural numbers having no prime factor in $\mathcal{P}$.

Question: Do we have an asymptotic formula for $\#\left(\mathcal{S} \cap [1, x]\right)$ ?

For $\kappa \in (0, 1/2)$, Corollary 1.2 of [1] gives that $$\#\left(\mathcal{S} \cap [1, x]\right) \sim C(\mathcal{P}) \frac{x}{(\log x)^\kappa} \quad (x \to +\infty)$$ where $C(\mathcal{P})$ is a positive constant depending on $\mathcal{P}$. I guess that something similar holds in the whole range $\kappa \in (0, 1)$, but I could not find such a result. Thank you for any help.

P.S. Although I am interested in general set of primes $\mathcal{P}$ of relative density $\kappa$, results concerning sufficiently general $\mathcal{P}$ of relative density $\kappa$ are welcome. For example, in light of Daniel Loughran's comment, sets like those of Chebotarev density theorem $$\mathcal{P} = \{p \text{ unramified in $K$}, \mathrm{Frob}_p \subseteq C\} ,$$ where $K$ is a finite Galois extension of $\mathbb{Q}$ and $C$ is a conjugacy class of its Galois group, are certainly interesting.

P.S. This is very related to question: Natural density of set of numbers not divisible by any prime in an infinite subset , as pointed out by user Hhhhhhhhhhh.

[1] Iwaniec and Kowalski, Analytic Number Theory (2004)

Answer 1

You basically ask about the sum $$ \sum_{n \le x} \alpha(n)$$ where $\alpha$ is a completely multiplicative function with $\alpha(p) = \mathbf{1}_{p \notin \mathcal{P}}$.

This is addressed by Wirsing in his famous paper ``Das asymptotische Verhalten von Summen über multiplikative Funktionen'' (Math. Ann. 143 (1961), 75–102). The only requirement on $E$ is $E(x)=o(\pi(x))$, and it gives the asymptotic result $$\tag{$\star$} \sum_{n \le x} \alpha(n) \sim\frac{ e^{-\gamma (1-\kappa)}}{\Gamma(1-\kappa)} \frac{x}{\log x} \prod_{p \le x,\, p \notin \mathcal{P}}(1-1/p)^{-1},$$ where $\gamma$ is the Euler-Mascheroni constant (appearing also in Mertens' theorem).

Remark 1: Suppose $\sum_{p \le x, p \in \mathcal{P}} 1/p = \kappa\sum_{p \le x} 1/p +C + o(1)$, which holds if $E(x)$ is small enough, say $O(x/\log^{1+\varepsilon} x)$ (by partial summation). Then $$C(\mathcal{P}) :=\frac{ 1}{\Gamma(1-\kappa)} \prod_{p \notin \mathcal{P}}(1-1/p)^{-1} \prod_{p}(1-1/p)^{1-\kappa}$$ converges and the last result may be simplified as $$C(\mathcal{P})\frac{x}{(\log x)^{\kappa}},$$ by Mertens. This should recover the result from Iwaniec and Kowalski.

Remark 2: In Wirsing's sequel to his own paper, ``Das asymptotische Verhalten von Summen über multiplikative Funktionen. II'' (Acta Math. Acad. Sci. Hungar. 18 (1967), 411–467) he relaxes the condition on $\mathcal{P}$ even further, requiring less than positive relative density, while still retaining $(\star)$.