Zeta functions of generalized permutations with application to their factorization formulas

Abstract

We obtain a determinant expression of the zeta function of a generalized permutation over a finite set. As a corollary we prove the functional equation for the zeta function. In view of absolute mathematics, this is an extension from $GL(n,\mathbf{F}_{1})$ to $GL(n,\mathbf{F}_{1^{m}})$, where $\mathbf{F}_{1}$ and $\mathbf{F}_{1^{m}}$ denote the imaginary objects “the field of one element” and “its extension of degree $m$”, respectively. As application we obtain a certain product formula for the zeta function, which is analogous to the factorization of the Dedekind zeta function into a product of Dirichlet $L$-functions for an abelian extention.