Geodesic
Asymptotic formula for the number of points of a~lattice in the circle on the Lobachevsky plane
Diskretnaya Matematika, Tome 18 (2006) no. 4, pp. 9-17.

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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 $N(z_0,T)$. 
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G. I. Arkhipov; V. N. Chubarikov. Asymptotic formula for the number of points of a~lattice in the circle on the Lobachevsky plane. Diskretnaya Matematika, Tome 18 (2006) no. 4, pp. 9-17. http://geodesic.mathdoc.fr/item/DM_2006_18_4_a1/

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