Let $f(x,y) = ax^2 + 2bxy + cy^2 \in \mathbb{Z}[x,y]$ be a positive definite binary quadratic form with even discriminant. We say that $f$ is *primitive* if $\gcd(a,2b,c) = 1$ and it is *reduced* if $0 < a < c$ and $|2b| \leq a$. Can we count, on average, those $f$ satisfying the above conditions and the condition that $\gcd(b,c) = 1$?

If we do not impose that $\gcd(b,c) = 1$, then the average count was done by Siegel. In particular he proved that

$$\displaystyle \sum_{N \leq x} h_{-4N} \sim \frac{4\pi}{21 \zeta(3)} x^{3/2}, $$

where $h_{-4N}$ is the number of reduced and primitive positive definite binary quadratic forms with discriminant $-4N$.

To more precisely formulate my question, put $h_{-4N}^\ast$ to be the number of reduced, primitive, and positive definite binary quadratic forms $f(x,y) = ax^2 + 2bxy + cy^2$ such that $N = ac - b^2$ and $\gcd(b,c) = 1$. Then can one give an asymptotic estimate for the quantity

$$\displaystyle \sum_{N \leq x} h_{-4N}^\ast?$$