Prime geodesic theorem

Closed geodesics on the surface SL(2,Z)\H, where H is the upper half plane, satisfy an asymptotic law that is very similar to the one describing the distribution of prime numbers. Moreover, the error term in this asymptotic law can be related to the spectrum of the Laplace operator, which is also the set of zeros of the Selberg zeta function. In this talk, I discuss how to estimate the error term and present a bound on its mean square. This is joint work with Giacomo Cherubini.