The average number of a class of reduced, primitive, positive definite binary quadratic forms

Question

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

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

Answer 1

Not an answer, but long for a comment. Siegel's theorem referenced above does not see the individual forms, it instead uses the Dirichlet class number formula for $L(1,\chi)$ and estimates the asymptotics of the sum.

If I were trying to prove the desired result, I'd first want a version of the proof that estimated the asymptotics of the residue of the sum over Epstein zeta functions, in order to see the individual forms. To pick out those with $\gcd(b,c)=1$, I would write this condition as $$ \sum_{d|\gcd(b,c)}\mu(d), $$ and then try to change the order of summation