Refined Counting of Geodesic Segments in the Hyperbolic Plane

For Γ a cofinite Fuchsian group, and l a fixed closed geodesic, we study the asymptotics of the number of those images of l that have a prescribed orientation and distance from l less than or equal to X. We give a new concrete proof of the error bound O(X2/3) that appears in the works of Good and Hejhal. Furthermore, we prove a new bound O(X1/2logX) for the mean square of the error. For particular arithmetic groups, we provide interpretations in terms of correlation sums of the number of ideals of norm at most X in associated number fields, generalizing previous examples due to Hejhal.