### Abstract

By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor-Selberg zeta function. It is shown that the Milnor-Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.

Original language | English |
---|---|

Pages (from-to) | 120-145 |

Number of pages | 26 |

Journal | Journal of Geometry and Physics |

Volume | 64 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology

### Cite this

*Journal of Geometry and Physics*,

*64*(1), 120-145. https://doi.org/10.1016/j.geomphys.2012.10.015

**Milnor-Selberg zeta functions and zeta regularizations.** / Kurokawa, Nobushige; Wakayama, Masato; Yamasaki, Yoshinori.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 64, no. 1, pp. 120-145. https://doi.org/10.1016/j.geomphys.2012.10.015

}

TY - JOUR

T1 - Milnor-Selberg zeta functions and zeta regularizations

AU - Kurokawa, Nobushige

AU - Wakayama, Masato

AU - Yamasaki, Yoshinori

PY - 2013/2/1

Y1 - 2013/2/1

N2 - By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor-Selberg zeta function. It is shown that the Milnor-Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.

AB - By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant expression of the Selberg zeta function, this higher depth determinant can be expressed as a product of multiple gamma functions and what we call a Milnor-Selberg zeta function. It is shown that the Milnor-Selberg zeta function admits an analytic continuation, a functional equation and, remarkably, has an Euler product.

UR - http://www.scopus.com/inward/record.url?scp=84875255423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875255423&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2012.10.015

DO - 10.1016/j.geomphys.2012.10.015

M3 - Article

VL - 64

SP - 120

EP - 145

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 1

ER -