Proceedings of the Japan Academy, Series A, Mathematical Sciences

Zeta functions of generalized permutations with application to their factorization formulas

Shin-ya Koyama and Sachiko Nakajima

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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.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 8 (2012), 115-120.

First available in Project Euclid: 4 October 2012

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Zentralblatt MATH identifier

Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Zeta functions the field with one element absolute mathematics generalized permutation groups


Koyama, Shin-ya; Nakajima, Sachiko. Zeta functions of generalized permutations with application to their factorization formulas. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 8, 115--120. doi:10.3792/pjaa.88.115.

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