Density of products of a certain set of primes

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime divisors in S. (That is, for each prime $p$ dividing such a number, $p\in S$.)

What are good methods for going about this? The trivial cases are when $S$ is finite or cofinite (in the set $\mathcal{P}$ of primes):

  • If $S$ is finite a Mertens-like product would give a relative density.
  • If $\mathcal{P}\setminus S$ is finite

Some particular cases of interest:

  • $S$ is the set of primes in a finite collection of arithmetic progressions: $S=\mathcal{P}\cap\left(\bigcup_{i=1}^k(a_i+b_in)_{n\in\mathbb{N}}\right)$.
  • $S$ is not known, but $s_n$, the n-th term of $S,$ obeys $f(n)\le s_n\le g(n)$ for sufficiently nice function $f,g.$ Example: $n^3\le s_n\le n^4$ for $n>100.$

These include very nice problems like “how common are numbers which are the sum of two squares” and “how dense are abundant numbers”.

An asymptotic would be ideal, but at this point I’ll take what I can get.

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