Meromorphicity of some deformed multivariable zeta functions for $F_1$-schemes

TL;DR

This paper introduces multivariable deformed zeta functions for $ ext{F}_1$-schemes, generalizing previous zeta functions, and proves their meromorphicity using explicit Hurwitz zeta function representations.

Contribution

It defines new multivariable deformed zeta functions for $ ext{F}_1$-schemes and establishes their meromorphicity through explicit formulas involving Hurwitz zeta functions.

Findings

Zeta functions are expressed explicitly via Hurwitz zeta functions.

The meromorphicity of these zeta functions is proven.

The invariant $ ext{ extmu}(A)$ is crucial in the analysis of Soulé type zeta functions.

Abstract

Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soul\'e type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivative of the modified Soul\'e type zeta functions, we find our invariant $\mu(A)$ for a finite abelian group $A$, introduced in ArXiv-0907.0918v2, plays an extremely important role in the Soul\'e type zeta functions.