TITLE: Does a positive binary quadratic form represent a set of primes possessing a natural density
QUESTION [9 upvotes]: In his answer to my question 
The Green-Tao theorem and positive binary quadratic forms
                Kevin Ventullo answers my initial question in the affirmative. What remains is the title question here, of separate interest to me. 
Any integral positive binary quadratic form integrally represents a set of primes with known Dirichlet density. This is an application of the Cebotarev density theorem (or Chebotarev or Tchebotarev), in particular it is Theorem 9.12 in David A. Cox, Primes of the form $x^2 + n y^2,$ with the example $\Delta = -56$ on page 190. I typed this out in the previous question.
Now, Jurgen Neukirch "Class Field Theory" points to Serre "A Course in Arithmetic," and on page 76 Serre says that the set of primes p such that a fixed polynomial has a root $\pmod p$ has a natural density, and refers to K. Prachar "Primzahlverteilung" chapter 5 section 7. By results (theorem 9.2, page 180) in the Cox book, this means that the principal form 
$x^2+ny^2$ or $x^2+xy+ky^2$ does represent a set of primes with a natural density, therefore equalling the Dirichlet density. And by the result on arithmetic progressions, a full genus of forms has a natural density.
Combining observations, the principal form always has a natural density of primes, any full genus does, therefore we are done for one class per genus, and in the case of two classes per genus we are done with the principal genus and any genus with two distinct opposite forms. So we have natural densities for Cox's $\Delta = -56$ example. We are also done with the principal genus when it has three classes.
So, (and I would love a reference), does every positive binary quadratic form represent a set of primes for which the natural density exists?

REPLY [6 votes]: If $Q$ is a positive binary quadratic form whose discriminant $D$ is fundamental, then the number of primes $\leq X$ represented by $Q$ is given asymptotically as
$\pi_{Q}(X)=\frac{1}{2h(D)} \mathrm{Li}(X)+O(X \exp{(-c_{Q}\sqrt{\log{X}})})$. 
Here $\mathrm{Li}(X)$ is the usual logarithmic integral. My reason for giving this as an answer is to point out that this was proven by de la Vallee Poussin himself, in the same paper where he proved the usual prime number theorem!  Dating from the 1890s, this definitely predates Chebotarev, and de la V.P.'s contribution deserves (IMHO) to be better known than it is.  Hadamard's work on the PNT is a little easier on the reader than de la V.P.'s, but de la V.P.'s results were both stronger and more general, which sometimes is forgotten.