Suppose $\mathbb P$ is a set of primes, and more specifically I'm interested in infinite $\mathbb P$ but satisfying \[ \sum _{p\in \mathbb P}\frac {1}{p}<\infty .\] How can I evaluate \[ \sum _{n\leq x\atop {p|n\implies p\in \mathbb P}}1\] and in particular has this been done already somewhere? If $\mathbb P$ has some size restriction, then it's possible to get asymptotics using sieve methods, but I want no such restriction.
The associated zeta function, if I've understood correctly, is the \emph {Burgess} zeta function \[ \zeta _\mathbb P(s)=\sum _{n=1\atop {p|n\implies p\in \mathbb P}}^\infty \frac {1}{n^s}=\prod _{p\in \mathbb P}(1-p^{-s})^{-1}.\] Now this may not have a meromorphic continuation (if I've understood some of the comments in https://arxiv.org/search/?query=A+Remark+on+Partial+Sums+Involving+the+Mobius+Functions&searchtype=all&source=header) but the question of the counting function above is much easier, right?