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Higher Selberg Zeta Functions

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Abstract

In the paper [KW2] we introduced a new type of Selberg zeta function for establishing a certain identity among the non-trivial zeroes of the Selberg zeta function and of the Riemann zeta function. We shall call this zeta function a higher Selberg zeta function. The purpose of this paper is to study the analytic properties of the higher Selberg zeta function z Γ(s), especially to obtain the functional equation. We also describe the gamma factor of z Γ(s) in terms of the triple sine function explicitly and, further, determine the complete higher Selberg zeta function with having a discussion of a certain generalized zeta regularization.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, Oh-okayama Meguro-ku, Tokyo, 152-8551, Japan

    Nobushige Kurokawa

  2. Faculty of Mathematics, Kyushu University, Hakozaki Higashi-ku, Fukuoka, 812-8581, Japan

    Masato Wakayama

Authors
  1. Nobushige Kurokawa
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  2. Masato Wakayama
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Correspondence to Masato Wakayama.

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P. Sarnak

Work in part supported by Grant-in Aid for Scientific Research (B) No.11440010, and by Grant-in Aid for Exploratory Research No.13874004, Japan Society for the Promotion of Science

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Kurokawa, N., Wakayama, M. Higher Selberg Zeta Functions. Commun. Math. Phys. 247, 447–466 (2004). https://doi.org/10.1007/s00220-004-1065-z

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