# Numbers made up of primes from a given set

Take a set $$\mathcal P$$ of primes and denote by $$\langle \mathcal P\rangle$$ the set of all natural numbers composed of primes from $$\mathcal P$$. If $\sum _{p\in \mathcal P}\frac {1}{p}$ converges then it can be shown without too much trouble that $\sum _{n\leq x\atop {n\in \langle \mathcal P\rangle }}1\ll x/\log x.$ Is is possible to get an asymptotic result for this count? I suspect that if we know something about the sizes of the elements of $$\mathcal P$$ then we can say something with sieve methods, but in this generality is it still possible to say something?

• I'd ask your question up front and then give context Oct 27 '21 at 17:03
• It seems likely that Theorem 4 in Pollack's paper pollack.uga.edu/HRmult5.pdf (A generalization of the Hardy--Ramanujan inequality and applications) could be applied (or an earlier result of Shirokov, cited there). If the set of primes has counting behaving like x/(log x)^{\beta} for a \beta > 1, there's something in Wirsing's 1961 paper: E. Wirsing, "Das asymptotische Verhalten von Summen über multiplikative Funktionen" Math. Ann. , 143 (1961) pp. 75–102 Oct 27 '21 at 19:31
• mathworker21 - ye, i've actually now taken out the context, not really necessary for the question (i think i just wanted to write out the argument cleanly for my own sake). so-called friend - cool! will have a look, thanks Oct 27 '21 at 19:54
• Please use a high-level tag like "nt.number-theory". I added this tag now. Oct 27 '21 at 20:05
• (I should have read the question more carefully: I don't want to claim that these references settle the question in full generality. But they should give an asymptotic formula for "most" naturally occurring examples of \P.) Oct 27 '21 at 20:47