The distribution of lattice points on the hyperboloid

Let $d$ be an integer. Denote by
$$ K_{\mathbb{Z}}(d)\,=\,\bigl\{(a,b,c)\in \mathbb{Z}^{3}\,\:\,b^{2}-4ac\,=\,d\bigr\} $$
the set of lattice points lying on the hyperboloid
$$ \bigl\{(x_{1},x_{2},x_{3})\in \mathbb{R}^{3}\,:\,x_{2}^{2}-4x_{1}x_{2}\,=\,d\bigr\}, $$
which is hyperbolic in the case $d<0$ and elliptic in the case $d>0$. Also, we denote
$$ K_{\mathbb{Z}}^{+}(d)\,=\,\bigl\{(a,b,c)\in K_{\mathbb{Z}}(d)\,\:\,c>0\bigr\}. $$
The set $K_{\mathbb{Z}}(d)$ is non-empty if and only if $d\equiv 0,1 \pmod 4$. Such numbers $d\ne 0$ are called as discriminants because the elements of the set $K_{\mathbb{Z}}(d)$ parametrize the binary quadratic forms $Q(u,v) = au^{2}+buv+cv^{2}$ of discriminant $d$ with integers coefficients. Next, let $\delta_{q}(m) = 1$ if $m\equiv 0 \pmod q$, and $\delta_{q}(m) = 0$ otherwise. Then Dirichlet series
$$ \sum\limits_{c=1}^{+\infty}\biggl(\,\sum\limits_{b \pmod{2c}}\delta_{4c}(b^{2}-d)\biggr)\frac{1}{c^{s}}\,=\,\frac{\zeta(s)}{\zeta(2s)}\,G_{d}(s)\quad (d\ne n^{2}) $$
converges absolutely in the half -plane $\Re s > 1$ and determines the entire function $G_{d}(s)$. The following theorem holds true:
Theorem. Suppose that $d\ne n^{2}$, $d\equiv 0,1 \pmod 4$, and let $\varphi(x,y)$ be any smooth complex-valued function over $\mathbb{R}\times (0,+\infty)$ with a compact support. Then, for any fixed $\varepsilon > 0$ we have}
$$ \sum\limits_{(a,b,c)\in K_{\mathbb{Z}}^{+}(d)}\varphi\biggl(\frac{b}{2c},\frac{\sqrt{|d|}}{2c}\biggr)\,= $$

$$ =\,\frac{3}{\pi^{2\mathstrut}}\,\sqrt{|d|}G_{d}(1)\int_{-\infty}^{+\infty}\int_{0}^{+\infty}\varphi(x,y)\,\frac{dx\,dy}{y^{2\mathstrut}}\,+\,O_{\varphi,\varepsilon}\bigl(|d|^{1/2-1/12+\varepsilon}\bigr). $$