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Spectral functions, special functions and the Selberg zeta function. (English) Zbl 0631.10025

The author starts from an arbitrary sequence (λ k ) k1 of positive real numbers such that λ k which is subject to suitable regularity conditions. (Typically, (λ k ) k1 will be the spectrum of a differential operator or a number-theoretically defined sequence.) Then he forms the associated θ-series, the Fredholm determinant, the zeta-function and the functional determinant and he discusses the relations between these functions by means of regularization techniques. In particular, he establishes a very general relation between the functional determinant and the Fredholm determinant.

The results apply to a wide variety of examples covering many classical situations from mathematics and theoretical physics. In particular, the spectral sequence of the Laplacian on the two-dimensional sphere is related with the Barnes G-function. The main application is an explicit factorization of the Selberg-zeta function into two functional determinants, one of which is expressible in terms of the Barnes G- function. Relations with some recent results on the Selberg zeta-function are also established.

Reviewer: J.Elstrodt

MSC:
11M35Hurwitz and Lerch zeta functions
35P99Spectral theory and eigenvalue problems for PD operators
58J50Spectral problems; spectral geometry; scattering theory
33B15Gamma, beta and polygamma functions
11F70Representation-theoretic methods in automorphic theory
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