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?

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    $\begingroup$ I'd ask your question up front and then give context $\endgroup$ Oct 27 '21 at 17:03
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    $\begingroup$ 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 $\endgroup$ Oct 27 '21 at 19:31
  • $\begingroup$ 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 $\endgroup$
    – tomos
    Oct 27 '21 at 19:54
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    $\begingroup$ Please use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$
    – GH from MO
    Oct 27 '21 at 20:05
  • $\begingroup$ (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.) $\endgroup$ Oct 27 '21 at 20:47

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