Monday, May 15, 2017

12:15–13:10

Schreiber 006

12:15–13:10

Schreiber 006

Anders Karlsson

Université de Genève & Uppsala University

Spectral zeta functions, graph limits and the Riemann Hypothesis

Following Carleman one forms zeta functions out of the spectrum of Laplacians, on manifolds and on graphs. For manifolds these functions have been studied and used in physics and topology in particular for defining determinants of Laplacians. In a joint work with F. Friedli, building on an earlier paper with G. Chinta and J. Jorgenson, 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-abstract 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 (GRH) 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 functional equations of Eisenstein series.