Does a positive binary quadratic form represent a set of primes possessing a natural density

Question

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?

Answer 1

Accoring to H. Lenstra, Chebotarev's theorem holds both for Dirichlet and for natural density (but he doesn't give a reference in this document). Applying Chebotarev to the extension $H/\mathbb{Q}$ where $H$ is the Hilbert class field of $\mathbb{Q}(\sqrt{-n})$ gives the result you want. (At least for primitive discriminants; for non-primitive discriminants you need an appropriate generalization of the Hilbert class field).

Added in response to Will's comment There is always a suitable field. Let $-D$ be a primitive negative discriminant and let $d=-m^2D$ be a general discriminant. Let $H_n$ be the abelian extension of $K=\mathbb{Q}{\sqrt{-D}}$ where a non-ramified prime splits iff its is principal and generated by an element of $\mathbb{Z}+m\mathcal{O}_K$. Such a field $H_n$ is called a ring class field and exists by class field theory. It also is an extension of $K$ by a singular value of the $j$-function.

Then $G_m=\mathrm{Gal}(H_n/\mathbb{Q})$ is a generalized dihedral group. There is a correspondence between conjugacy classes in $G_m$ and pairs of equivalence classes of $ax^2\pm bxy+cy^2$ of discriminant $d$, such that an unramified prime has its Frobenius in a conjugacy class iff it it represented by the corresponding form. This is why we can apply Chebotarev.

Even more added A good reference for ring class fields is Cox's book Primes of the Form $x^2+ny^2$.

Answer 2

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.