, Volume 247, Issue 2, pp 447-466

Higher Selberg Zeta Functions

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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.

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