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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!

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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).

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  • $\begingroup$ I'll accept the answer, though I realize now that what I was really looking for was this constant $C$ explicitly. I couldn't find this exact derivation in Landau.. $\endgroup$
    – Tian An
    Commented Apr 2, 2017 at 16:58
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    $\begingroup$ It is quite easy to write down the constant $C$ explicitly. One can write the associated Dirichlet series in terms of Dirichlet $L$-functions. One then obtains the asymptotic formula and thus $C$ via a Tauberian theorem ($C$ will be written in terms of special values of Dirichlet $L$-functions, which can be "calculated" using the class number formula). These methods are all explained very well in the cited paper of Serre; I would very much recommend that you study it. $\endgroup$
    – Daniel Loughran
    Commented Apr 2, 2017 at 18:03
  • $\begingroup$ The constant in the closely related case of representations of integers by binary quadratic forms can be found in: Brink, Moree, Osburn - Principal forms $x^2+ny^2$ representing many integers. $\endgroup$
    – Daniel Loughran
    Commented Apr 2, 2017 at 18:05

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