The Transfer Operator Approach to Selberg’s Zeta Function and Modular and Maass Wave Forms For P S L (2, ℤ)

Abstract

In this paper we discuss the transfer operator approach to Selberg’s zeta function for P S L(2, ℤ). Since this function can be expressed as the Fredholm determinant det(1 - L β ) of the transfer operator L β , β ∈ C for the geodesic flow on the modular surface, the zeros and poles of the Selberg function are closely related to those β-values, where L β has an eigenvalue λ = 1 respectively where L β has poles. It turns out that the corresponding eigenfunctions of L β for eigenvalues λ = 1 are closely related to both holomorphic and non-holomorphic modular forms respectively the Maass wave forms. Therefore these eigenfunctions, which by definition of L β are holomorphic functions, are by themselves interesting quantities for the group P S L (2, ℤ): indeed special cases are the period polynomials and functions of the Manin-Eichler and Shimura theory of periods for this group. Another special example of such an eigenfunction is the well known density of Gauss’s measure for the continued fraction expansion. The transfer operator approach hence in a surprising way combines several aspects of the theory of modular and Maass wave forms for the modular group, which up to now were not directly related.Key wordsquantum chaosSelberg zeta functiondynamical zeta functiontransfer operatorfunctional equationmodular formsMaass wave formsperiod polynomialperiod function