Spectral functions, special functions and the Selberg zeta function (Q1094445)

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.