Spectral theory of graphs and Riemann’s zeta function in the critical strip

We define zeta functions, different from Ihara’s, associated to the Laplace spectrum of finite and infinite graphs. In asymptotics, zeta functions from number theory appear. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of explicit graph zetas. Some graph zetas appear moreover already in classical analytic number theory. Special values are considered and certain hypergeometric functions arise.

Joint work with Fabien Friedli.