Systolic zeta function

Some homological length spectrum is defined for every Riemannian manifold. Such a spectrum consists of the lengths of the shortest closed geodesics in each 1-dimensional integer homology class of the manifold. A Dirichlet type generating function of such a spectrum is introduced in the talk. Some connections of the analytic properties of such a function with topological and metric properties of a given manifold are discussed in the talk as well.