Abstract
We use B. Randol’s method to improve the error term in the prime geodesic theorem for a noncompact Riemann surface having at least one cusp. The case considered is a general one, corresponding to a Fuchsian group of the first kind and a multiplier system with a weight on it.
Highlights
PreliminariesLet X be a non-compact Riemann surface regarded as a quotient Γ \ of the upper half-plane by a finitely-generated Fuchsian group Γ ⊆ PSL (2, ) of the first kind, containing n1 ≥ 1 cusps
Open Access the hyperbolic Laplacian in terms of geometric data involving the lengths of geodesics on a Riemann surface
Motivated by analogy between this trace formula and the explicit formulas of number theory relating the zeroes of the Riemann zeta function to prime numbers, Selberg [1] introduced a zeta function whose analytic properties are encoded in the Selberg trace formula
Summary
Let X be a non-compact Riemann surface regarded as a quotient Γ \ of the upper half-plane by a finitely-generated Fuchsian group Γ ⊆ PSL (2, ) of the first kind, containing n1 ≥ 1 cusps. Let I denote the fundamental region of Γ. We shall assume that the fundamental region I of Γ has a finite non-Euclidean area I. For an r dimensional vector space V over we consider an essentially self-adjoint operator. The operator −∆m has the unique self-adjoint extension −∆ m to the space m , a dense subspace of. By mj we will denote the multiplicity of 1 as an eigen-value of the matrix ( ) n1. The discrete spectrum will be denoted as. Gušić zeros (or equivalently poles) of the hyperbolic scattering determinant (see, [12])
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
