On systolic zeta functions (Q1695700)

Using the homological length specrum of a compact Riemannian manifold \(M\) with first Betti number \(b_1=b_1(M)\) the \textit{systolic zeta function} \(\zeta_{\mathrm{sys}}(z)\) is defined. This function is holomorphic in the half plane \(Re(z)>b_1\). It extends analytically to the half plane \( Re(z) < b_1-1\) with a simple pole at \(z=b_1.\) Then an inequality analogous to \textit{M. Gromov}'s isosystolic inequality [J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)]) is derived. Instead of the systole the residue \(\mathrm{Res}_{b_1}(\zeta_{\mathrm{sys}}(z))\) at \(b_1\) of \(\zeta_{\mathrm{sys}}(z)\) enters.