Last edited by Grom
Friday, July 31, 2020 | History

3 edition of Selberg zeta and theta functions found in the catalog.

Selberg zeta and theta functions

Ulrich Bunke

# Selberg zeta and theta functions

## by Ulrich Bunke

Written in English

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

• 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

M. Möller, A. Pohl: Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant; Erg. Th. Dyn. Syst. () [92] T. Mühlenbruch: Systems of Automorphic Forms and Period Functions; Ph.D. thesis Utrecht, He wrote the book An Introduction to the Riemann Zeta Function (Cambridge, ). Subjects that Patterson deals with include discontinuous groups (Fuchsian groups), different zeta functions (for example those of Ruelle and Selberg), metaplectic groups, generalized theta functions, and exponential sums in analytical number theory.

Selberg zeta function, which is a meromorphic function of a complex variable sthat is deﬂned for 1 in terms of the length spectrum and that has zeros at all s2C for which s(1¡s) is an eigenvalue of the Laplacian in L2(X). We will be interested in the case when Xis the quotient of the upper half-plane H by either the modular group. Browse other questions tagged -theory probability-distributions riemann-zeta-function or ask your own question. Featured on Meta Improved experience for users with review suspensions.

A generalisation of the Selberg zeta function, or rather its log-arithmic derivative, to higher rank is given. In the compact case one gets a Mittag-Leffler series expansion, the proof of which. Zeta and \(L\)-functions play a central role in number theory. They provide important information of arithmetic nature. This book, which grew out of the author's teaching over several years, explores the interaction between number theory and combinatorics using zeta and \(L\)-functions as a central theme.

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### Selberg zeta and theta functions by Ulrich Bunke Download PDF EPUB FB2

Also describes the theta function of Riemannian surfaces with cusps, and alternative descriptions of the singularities of the Selberg zeta functions in terms of Lie algebra and group cohomology.

Includes explicit worked examples. Distributed by VCH. Annotation copyright Book News, Inc. Portland, by:   The connection between these functions and the spectrum of certain elliptic differential operators is provided by a version of the Selberg trace formula.

The theta function is a regularized trace of the wave group. Originally defined geometrically, the Selberg zeta function has a representation in terms of regularized : Ulrich Bunke, Martin Olbrich.

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. 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, Format: Paperback.

The Notes give a direct approach to the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) acting on the upper half-plane. The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic Laplacian in order to arrive at the logarithmic derivative of the Selberg zeta-function.

Zeta functions and chaos, by Audrey Terras, PDF file. Vertex operators and modular forms, by Geoffrey Mason and Michael Tuite, PDF file.

Research Lectures Applications of elliptic and theta functions to FRLW cosmology with cosmological. The Selberg zeta-function was introduced by Atle Selberg ().It is analogous to the famous Riemann zeta function = ∏ ∈ − −where is the set of prime numbers.

The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers. If is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows, = ∏ (− −) −.

In [5], authors derived the properties of certain zeta functions canonically associated with the geodesic flow of Y. Motivated by the fact that the classical Selberg zeta function [17] is an.

Ebook Riemann's Zeta Function, by Harold M. Edwards. We will show you the most effective and most convenient way to get book Riemann's Zeta Function, By Harold M.

Edwards in this world. Great deals of compilations that will assist your task will certainly be here. 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.This book provides an introduction, with applications, to three interconnected mathematical topics: • zeta functions in their rich variety: those of Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle, plus graph zeta functions; • modular forms: Eisenstein series, Hecke and Dirichlet L-functions, Ra-manujan’s tau function, and cusp forms.

Moreover, introducing a “poly-Selberg zeta function” Z Γ (m) (s) by a certain Euler product (which is regarded as another generalization of the Selberg zeta function), we show that Z Γ, r (s) can be expressed as a product and quotient of Z Γ (m) (s) (Theorem ).

This is nothing but the Euler product expression of Z Γ, r (s).