Motivic zeta functions and infinite cyclic covers

Abstract

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) a rational function in $K_0({\rm Var}^{\hat \mu}_{\mathbb{C}})[\mathbb{L}^{-1}]$, which we call {\it motivic infinite cyclic zeta function}, and show its birational invariance. Our construction is a natural extension of the notion of {\it motivic infinite cyclic covers} introduced by the authors, and as such, it generalizes the Denef-Loeser motivic Milnor zeta function of a complex hypersurface singularity germ.