Counting and zeta functions over F1

TL;DR

The paper introduces a novel interpretation of zeta functions for F1-schemes, extending their applicability beyond previous conditions, and connects F1-theory with spectral theory through regularization techniques.

Contribution

It provides a new framework for defining zeta functions for F1-schemes without Soulé's condition, linking algebraic geometry with spectral analysis.

Findings

Functional equations for reductive groups are derived.

A new regularization-based definition of zeta functions is proposed.

The approach unifies F1-theory with spectral theory of Laplace operators.

Abstract

A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soul\'e's condition. Functional equations for reductive groups are computed and a new definition of zeta functions attached to more general counting functions is given which is based on regularization and puts on an equal footing F1-theory on the one hand and spectral theory of Laplace operators on manifolds on the other.