Windings of prime geodesics

Abstract

The winding of a closed oriented geodesic around the cusp of the modular orbifold is computed by the Rademacher symbol, a classical function from the theory of modular forms. For a general cusped hyperbolic orbifold, we have a procedure to associate to each cusp a Rademacher symbol. In this paper we construct winding numbers related to these Rademacher symbols. In cases where the two functions coincide, access to the spectral theory of automorphic forms yields statistical results on the distribution of closed (primitive) oriented geodesics with respect to their winding.