Anders Karlsson: Graph zeta functional equations and the Riemann hypothesis

Following Carleman one forms zeta functions out of the spectrum of
Laplacians, on manifolds and on graphs. In a joint work with F.
Friedli we determine the asymptotics of the zeta function in certain
families of graphs and relate them to certain number theoretical
zetas. It turns out that in a non-trivial way a hypothetical
functional relation, of the type s vs 1-s, on the graph side is
equivalent to the Riemann hypothesis. Friedli showed moreover that
this picture persists when introducing a Dirichlet character on the
graph side, concerning the Riemann hypothesis for the corresponding
Dirichlet L-functions. The zeta function of the line graph Z is an
interesting function in itself, with a functional equation of the
standard type, s vs 1-s, extending the ubiquitous Catalan numbers in
combinatorics and appearing in the scattering determinant of
Eisenstein series.