In 1956, Selberg introduced a zeta function Z(s) defined in terms of closed geodesics on a negatively curved manifold, defined by analogy with the Riemann zeta function in number theory. In the case of manifolds with constant negative sectional curvatures the Selberg trace formula gives an extension of Z(s) to the entire complex plane. Using a more dynamical approach, Giulietti, Liverani and Pollicot have extended this result to the case of manifolds with variable negative sectional curvatures (and, moregenerally, Anosov flows).
Seminar flyer: Zeta Functions for Closed Geodesics
Originally published at math.nd.edu.