Asymptotic formula for the number of points of a lattice in the circle on the Lobachevsky plane

We define the distance $d=d(z,z')$ between points $z=x+iy$ and $z'=x'+iy'$ in the upper half-plane, setting
$$ d=\ln\biggl(\frac{u+2+\sqrt{u^2+4u}}2\biggr), $$
where
$$ u=\frac{|z-z'|^2}{yy'}\,. $$
The circle $K(z_0,T)$ with centre in a point $z_0$ consists of the points $z$ satisfying the inequality $d(z,z_0)\leq T$. Let $N(z_0,T)$ be the number of elements $\gamma$ of the modular group $\mathit{PSL}_2(\mathbf Z)$ such that the point $\gamma z_0$ lies in the circle $K(z_0,T)$. In this paper, we refine the remainder term in the asymptotic formula for $N(z_0,T)$.