Let $Q$ be an indefinite binary quadratic form with discriminant $D$ and one class per genus (keep the example $x^2 - 2y^2$ in mind). If one asks about the set $P = \{ p : p \text{ prime and } p = Q(x,y) \text{ for some } x,y \in \mathbb{Z} \}$. It's pretty easy to determine the relative density of $P$ in the set of all primes. However, suppose that we look at the set $P'(B) = \{ p : p \text{ prime and } p = Q(x,y) \text{ for some } x,y \in \mathbb{Z}, |x|,|y| \le B \}$ of primes represented by $Q$ where $(x,y)$ is in a box. Is it still true that $\lim_{B \rightarrow \infty} |P'(B)|/\pi(C(B)) = \lim_{N \rightarrow \infty} |\{ p \in P, |p| \le N \}|/\pi(N)$, where $C(B) = \max_{|x|,|y| \le B} |Q(x,y)|$?

I'm not completely sure of what should be the right denominator in the first limit. Perhaps one should look at $P'(B,N) = \{ p : p \text{ prime and } p = Q(x,y) \text{ for some } x,y \in \mathbb{Z}, |x|,|y| \le B, p \le N\}$ and have $B$ be a function of $N$, and ask for which functions the two limits are equal.