Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4.
Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up to $x$ for a constant $k\approx0.76422.$ Restricting to the squarefree members reduces the incidence by a factor of $\zeta(2).$
So there are $\sim0.46459\ldots x/\sqrt{\log x}$ products of primes =1,2 mod 4 up to $x$. Does this generalize to other congruence classes of primes? That is, given $m$ and some set $S$ and numbers which are the product of distinct primes $\equiv s\pmod m$ for some $s\in S$ (with at least one $s$ coprime to $m$), is their density $kx/\sqrt{\log x}$ for some suitable constant $k$?
Bonus: Is there a good way to find the constant given $m$ and $S$?