Density of primes of the form $x^2+my^2$

Question

I was playing with numbers and have the nice conjectures:

Let $m$ be a fixed positive integer, and $\pi(N)$ denote the numbers of primes not exceeding $N$ and $\pi_m(N)$ denote the number of prime not exceeding $N$ which are of the form $x^2+my^2$. Then

(a) $g_m := \displaystyle \lim_{N \rightarrow \infty} \frac{\pi(N)}{\pi_m(N)}$ exists and is always an nonzero even number.
(b) For an even number $2k$, let $f: 2 \mathbb{Z} \mapsto 2^{\mathbb{Z}}$ scuh that $f(2k)$ be the number of integers such that $g_m = k$. Then $|f(2k)|$ is finite and nonzero for all $k$

I have no idea how should I even have an attempt at proving this. Where should I start ?

Answer 1

yes, this is Theorem 9.12 in Primes of the Form $x^2 + n y^2$ by David A. Cox. Indeed, the ratio is $2 h(-4m), $ where the $h$ refers to the form class number of that discriminant.

As far as your part (b), the class number goes to infinity as $m$ increases. Slowly.

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