On Systolic Zeta Functions

TL;DR

This paper introduces Dirichlet series linked to homology length spectra of manifolds and polyhedra, deriving inequalities that extend Gromov's intersystolic inequality to the entire spectrum.

Contribution

It defines new Dirichlet series for homology length spectra and establishes analytical properties and inequalities involving these series.

Findings

Derived an inequality extending Gromov's intersystolic inequality

Established analytical properties of the new Dirichlet series

Connected homology length spectra to spectral inequalities

Abstract

We define Dirichlet type series associated with homology length spectra of Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their analytical properties. As a consequence we obtain an inequality analogous to Gromov's classical intersystolic inequality, but taking the whole homology length spectrum into account rather than just the systole.