# Motivic zeta functions and infinite cyclic covers

## TL;DR

This paper introduces a new motivic infinite cyclic zeta function associated with infinite cyclic covers of complex manifolds, extending previous concepts and demonstrating its birational invariance.

## Contribution

It constructs a birational invariant motivic zeta function for infinite cyclic covers, generalizing the motivic Milnor zeta function for hypersurface singularities.

## Key findings

- Defines the motivic infinite cyclic zeta function as a rational function.
- Shows the birational invariance of the zeta function.
- Generalizes the motivic Milnor zeta function for complex hypersurface singularities.

## 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.