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

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.