We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions $η(δ_t,s)$ and $tζ(δ_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms.
Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion
Adiabatic decomposition of the zeta-determinant of the Dirac Laplacian I. The case of an invertible tangential operator. With an appendix by Yoonweon Lee
Dynamical Zeta Functions, Nielsen theory and Reidemeister torsion
The paper consists of four parts. Part I presents a brief survey of the Nielsen fixed point theory. Part II deals with dynamical zeta functions connected with Nielsen fixed point theory. Part III is concerned with congruences for the Reidemeister and Nielsen numbers. Part IV deals with the Reidemeister torsion . In Chapter 2 we prove that the Reidemeister zeta function of a group endomorphism is a rational function with functional equation in the following cases: the group is finitely generated and an endomorphism is eventually commutative; the group is finite ; the group is a direct sum of a …
