On the number of lattice points in a circle on the Lobachevskij plane (Q1345664)

In order to estimate the number of lattice points in a circle on the Lobachevskij plane one has to investigate the following lattice point problem on the Euclidean space. Let \(2N(T)\) be the number of solutions of \(ad - bc = 1\), \(a^2 + b^2 + c^2 + d^2\leq 2 \text{cosh} T\), where \(a,b,c,d\) are integers. It is proved that \[ N(T)=6 \text{cosh} T+O(T^3e^{3T/4}). \]