Windings of prime geodesics

In joint work with Flemming von Essen, we use modular forms to construct a function that measures the winding of closed geodesics about a prescribed cusp of a hyperbolic surface. On the modular surface, this winding number is given by the Rademacher function $\psi$, which is a conjugacy class invariant on the modular group, arises from the study of the multiplier system of Dedekind’s eta function, and computes hyperbolic periods of Hecke’s weight 2 Eisenstein series. We obtain the following Dirichlet-type equidistribution theorem for various families of surfaces: Given a set $A$ of integers with natural density $d(A)$, the set of prime geodesics with winding number in $A$ has density $1/d$.