where the product is taken over
all primitive closed geodesics
with
denoting their length.
Remarkably, this zeta function
continues meromorphically to all of
.
Using recent advances in the study of
resonances for Anosov flows and simple arguments
from microlocal analysis, we prove that for an
orientable negatively curved surface,
the order of vanishing of
at
is given by
the absolute value of the Euler
characteristic. In constant curvature this
follows from the Selberg trace formula and this is the
first result of this kind for manifolds which are
not locally symmetric. This talk is based on
joint work with Maciej Zworski.
.