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Spectral functions, special functions and the Selberg zeta function. (English) Zbl 0631.10025
The author starts from an arbitrary sequence \((\lambda_k)_{k\geq 1}\) of positive real numbers such that \(\lambda_k\to \infty\) which is subject to suitable regularity conditions. (Typically, \((\lambda_k)_{k\geq 1}\) 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.
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.

MSC:
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
35P99 Spectral theory and eigenvalue problems for partial differential equations
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J52 Determinants and determinant bundles, analytic torsion
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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