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Zbl 0631.10025
Voros, A.
Spectral functions, special functions and the Selberg zeta function. (English)
[J] Commun. Math. Phys. 110, 439-465 (1987). ISSN 0010-3616; ISSN 1432-0916/e

The author starts from an arbitrary sequence $(\lambda\sb k)\sb{k\ge 1}$ of positive real numbers such that $\lambda\sb k\to \infty$ which is subject to suitable regularity conditions. (Typically, $(\lambda\sb k)\sb{k\ge 1}$ will be the spectrum of a differential operator or a number-theoretically defined sequence.) Then he forms the associated $\theta$-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. \par 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.
[J.Elstrodt]

MSC 2000:: *11M35 Other zeta functions
35P99 Spectral theory and eigenvalue problems for PD operators
58J50 Spectral problems; spectral geometry; scattering theory
33B15 Gamma-functions, etc.
11F70 Representation-theoretic methods in automorphic theory

Keywords: Selberg trace formula; compact Riemann surface; Glaisher-Kinkelin constant; Fredholm determinant; zeta-function; functional determinant; spectral sequence; Laplacian; Barnes G-function; factorization of the Selberg-zeta function

Cited in: Zbl 1090.11053 Zbl 0844.11037 Zbl 0708.11039 Zbl 0641.33003 Zbl 0900.11032

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Zentralblatt MATH Berlin [Germany]