Proceedings of the Japan Academy, Series A, Mathematical Sciences

The Riemann hypothesis and functional equations for zeta functions over ${\bf F}_1}$

Sojung Kim, Shin-ya Koyama, and Nobushige Kurokawa

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Abstract

We prove functional equations for the absolute zeta functions. We also show that the absolute zeta functions satisfy the tensor structure in the sense that their singularities possess an additive property under the tensor product. Moreover those singularities satisfy the analog of the Riemann hypothesis.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci. Volume 85, Number 6 (2009), 75-80.

Dates
First available in Project Euclid: 3 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.pja/1244037801

Digital Object Identifier
doi:10.3792/pjaa.85.75

Mathematical Reviews number (MathSciNet)
MR2532423

Subjects
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}

Keywords
Zeta functions the field with one element absolute mathematics

Citation

Kim, Sojung; Koyama, Shin-ya; Kurokawa, Nobushige. The Riemann hypothesis and functional equations for zeta functions over ${\bf F}_1}$. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 6, 75--80. doi:10.3792/pjaa.85.75. https://projecteuclid.org/euclid.pja/1244037801.


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References

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