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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)

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  • $\begingroup$ Do you have a specific application in mind? Usually the set $\mathcal{P}$ arises in some natural way, e.g. as the set of primes which have some splitting property in some number field. In such cases one can usually get asymptotics using Dirichlet series techniques. $\endgroup$ Sep 2, 2021 at 8:54
  • $\begingroup$ @DanielLoughran I had in mind general sets, but I think that also the particular cases you mention might be interesting (at least to understand what is possible at best). I added a postscriptum, thanks $\endgroup$
    – annie
    Sep 2, 2021 at 9:15
  • $\begingroup$ Probably this link might help. mathoverflow.net/q/369092/160943 $\endgroup$ Sep 2, 2021 at 10:07

1 Answer 1

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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 $(\star)$.

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