In his 2006 ICM lecture, Ghys observed that the winding of a closed geodesic around the cusp of the modular surface can be computed using a function from number theory: the Rademacher function. In a 2007 letter, Sarnak sketched how to use this connection to obtain statistical information on the windings of modular geodesics. In recent work, I study how and when Rademacher functions encode the winding of closed geodesics around a prescribed cusp of an arbitrary cusped hyperbolic surface and what statistical information can be deduced. In this talk, I will describe these winding numbers, give an overview of the current state of knowledge, and discuss some arising open questions.