Selberg zeta and theta functions
Share

# Selberg zeta and theta functions a differential operator approach by Ulrich Bunke

• 29 Want to read
• ·
• 44 Currently reading

Published by Akademie Verlag, VCH in Berlin, New York .
Written in English

### Subjects:

• Selberg trace formula.,
• Functions, Theta.,
• Functions, Zeta.

## Book details:

Edition Notes

Includes bibliographical references (p. [159]-165) and index.

Classifications The Physical Object Statement Ulrich Bunke, Martin Olbrich. Series Mathematical research,, v. 83, Mathematical research ;, Bd. 83. Contributions Olbrich, Martin, 1963- LC Classifications QA241 .B825 1995 Pagination 168 p. : Number of Pages 168 Open Library OL1273168M ISBN 10 3055016904 LC Control Number 95004564

### Download Selberg zeta and theta functions

PDF EPUB FB2 MOBI RTF

A comprehensive exposition of the theory of Selberg zeta and theta functions for bundles on compact locally symmetric spaces of rank 1. The connection between these functions and the spectrum of certain elliptic differential operators is provided by a version of the Selberg trace formula. This book presents a method for evaluating Selberg zeta functions via transfer operators for the full modular group and its congruence subgroups with characters. Studying zeros of Selberg zeta functions for character deformations allows us to access the discrete spectra and resonances of hyperbolic Laplacians under both singular and non. In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + it).It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. "Selberg noticed this similarity and was quickly led to a deeper study of trace formulas. Among other things, Selberg found that there is a zeta function which corresponds to [his trace formula] in the same way that [the Riemann zeta function] corresponds to [the Riemann-Weil explicit formula].This zeta function is nowadays referred to as the Selberg zeta function; it is usually denoted by Z.

This book provides an introduction to (1) various zeta functions (for example, Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle), including graph zeta functions; (2) modular forms (Eisenstein series, Hecke and Dirichlet L-functions, Ramanujan's tau function, and cusp forms); and (3) vertex operator algebras (correlation functions, quasimodular forms, modular invariance, rationality, and. This book presents a method for evaluating Selberg zeta functions via transfer operators for the full modular group and its congruence subgroups with characters. Studying zeros of Selberg zeta functions for character deformations allows us to access the discrete spectra and resonances of hyperbolic. These are functions of a complex variable s in some right half-plane of \(\mathbb {C}\). Using the Selberg trace formula for arbitrary finite dimensional representations of the fundamental group of the manifold, we establish the meromorphic continuation of the dynamical zeta functions . Expression of the Selberg/Ruelle zeta function as an Euler product of characteristic polynomials is due to. Ramesh Gangolli, Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Illinois J. Math. Vol Issue 1 (), David Fried, The zeta functions of.

Selberg Zeta and Theta Functions. A Differential Operator Approach: 83 (Mathematical Research). Find all books from Ulrich Bunke, Martin Olbrich. At you can find used, antique and new books, compare results and immediately purchase your selection at the best price. This. Theta and Selberg Zeta Function Ulrich Bunke J Abstract Contents 1 The Theta function 1 2 Hyperbolic space 5 3 The Selberg Trace Formula 8. The trace formula -- 2. Theta Functions. Theta functions for compact locally symmetric rank-one spaces. Theta functions for surfaces with cusps -- 3. The Selberg Zeta Function. Zeta functions and the geodesic flow. Traces of resolvents. Properties of the Selberg zeta function -- 4. The Ruelle Zeta Function. The product converges to a non-zero analytic function for Re(s) >1 and Selberg showed the following fundamental result. Theorem (Selberg). The Selberg zeta function S(s;ˆ) extends to an entire function of order 2 and has a simple zero at s= 1. An account of this theorem is contained in Hejhal’s book [11], where it appears.