By a theorem of Landau, the number of integers $n\leq x$ whose prime divisors belong to only arithmetic progressions $a\_1,\dots,a\_r$ mod $q$, with $r\leq\varphi(q)$ and $a\_i$ coprime to $q$ for each $i$, is


$$Cx (\log x)^{r/\varphi(q)-1} + O(x (\log x)^{r/\varphi(q)-2}),$$


for some constant $C$ depending on $r$ and $q$.


This is an old result. I am sure much more is known about the distribution of such $n$, but I do not know the literature well enough, and would be happy if someone could provide suggestions regarding the state of the art (or other classical theorems) regarding results of this nature. Thanks!



Density of numbers whose prime factors belong to given arithmetic progressions
This result has been generalised a fair bit. The main generalisation is that you can replace congruence conditions by so-called "Frobenian conditions", namely conditions of the type which arise in the Chebotarev density theorem.
There have also been some improvements in the error term, but substantial improvements are not really possible without assuming GRH.


Serre has written quite a bit about such topics. See for example Theorem 2.8 of the paper:


Serre - Divisibilité de certaines fonctions arithmétiques.


The associated zeta functions don't admit a meromorphic continuation to all of $\mathbb{C}$ in general; they have natural boundaries along the line $\mathrm{re}(s) = 0$. You can read more about this in the paper:


Hashimoto - Partial zeta functions.


(the author works in an even greater generality than I describe here).


