CONTEMPORARY MATHEMATICS

290

Dynamical/ Spectral/ and Arithmetic Zeta Functions

AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions

January 15-16, 1999 San Antonio, Texas

Michel L. Lapidus Machiel van Frankenhuysen

Editors

Selected Titles in This Series

290 Michel L. Lapidus and Machiel van Frankenhuysen, Editors, Dynamical, spectral, and arithmetic zeta functions, 2001

289 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, Second summer school in analysis and mathematical physics: Topics in analysis: Harmonic, complex, nonlinear and quantization, 2001

288 Marisa Fernandez and Joseph A. Wolf, Editors, Global differential geometry: The mathematical legacy of Alfred Gray, 2001

287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001

286 Edward L. Green, Serkan Ho§ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001

285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001

284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001

283 Paul A. Milewski, Leslie M. Smith, Fabian Waleffe, and Esteban G. Tabak, Editors, Advances in wave interaction and turbulence, 2001

282 Arlan Ramsay and Jean Renault, Editors, Groupoids in analysis, geometry, and physics, 2001

281 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering II, 2001

280 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering I, 2001

279 Alejandro Adem, Gunnar Carlsson, and Ralph Cohen, Editors, Topology, geometry, and algebra: Interactions and new directions, 2001

278 Eric Todd Quinto, Leon Ehrenpreis, Adel Faridani, Fulton Gonzalez, and Eric Grinberg, Editors, Radon transforms and tomography, 2001

277 Luca Capogna and Loredana Lanzani, Editors, Harmonic analysis and boundary value problems, 2001

276 Emma Previato, Editor, Advances in algebraic geometry motivated by physics, 2001 275 Alfred G. Noel, Earl Barnes, and Sonya A. F. Stephens, Editors, Council for

African American researchers in the mathematical sciences: Volume III, 2001 274 Ken-ichi Maruyama and John W. Rutter, Editors, Groups of homotopy

self-equivalences and related topics, 2001 273 A. V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, and C. Vinsonhaler,

Editors, Abelian groups, rings and modules, 2001 272 Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki, Editors, Quadratic

forms and their applications, 2000 271 J.P. C. Greenlees, Robert R. Bruner, and Nicholas Kuhn, Editors, Homotopy

methods in algebraic topology, 2001 270 Jan Denef, Leonard Lipschitz, Thanases Pheidas, and Jan Van Geel, Editors,

Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000 269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations

and foliations in dynamics, geometry and topology, 2001 268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential

geometric methods in the control of partial differential equations, 2000 267 Nicolas Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen

Schneider, Editors, New trends in Hopf algebra theory, 2000 (Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/290

Dynamical, Spectral, and Arithmetic Zeta Functions

CoNTEMPORARY MATHEMATICS

290

Dynamical, Spectral, and Arithmetic Zeta Functions

AMS Special Session on DynamicaL Spectral, and Arithmetic Zeta Functions

January 15-16, 1999 San Antonio, Texas

Michel L. Lapidus Machiel van Frankenhuysen

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeThrck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume contains the proceedings of the AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions held at the Annual Meeting of the American Mathematical Society in San Antonio, Texas on January 15-16, 1999.

2000 Mathematics Subject Classification. Primary 11F67, llMxx, 11Y35, 11N05, 28A80, 30F40, 37 Axx, 58J35.

Library of Congress Cataloging-in-Publication Data AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions (1999: San Antonio, Texas)

Dynamical, spectral, and arithmetic zeta functions: AMS Special Session on Dynamical, Spec-tral, and Arithmetic Zeta Functions, January 15-16, 1999, San Antonio, Texas / Michel L. Lapidus, Machiel van Frankenhuysen, editors.

p. em. ~(Contemporary mathematics, ISSN 0271-4132; 290) Includes bibliographical references. ISBN 0-8218-2079-6 (alk. paper) 1. Functions, Zeta~Congresses. I. Lapidus, MichelL. (Michel Laurent), 1956- II. van Frank-

enhuysen, Machiel, 1967- III. Title. IV. Contemporary mathematics (American Mathematical Society) ; v. 290.

QA351.A73 1999 515'.56~dc21 2001053944

Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission\Dams. org.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

© 2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

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Contents

Preface

Eigenfunctions of the transfer operators and the period functions for modular groups

ix

CHENG-HUNG CHANG AND DIETER H. MAYER 1

A note on dynamical trace formulas CHRISTOPHER DENINGER AND WILHELM SINGHOF 41

Small eigenvalues and Hausdorff dimension of sequences of hyperbolic three-manifolds CAROL E. FAN AND JAY JORGENSON 57

Dynamical zeta functions and asymptotic expansions in Nielsen theory ALEXANDER FEL'SHTYN 67

Computing the Riemann zeta function by numerical quadrature WILLIAM F. GALWAY 81

On Riemann's zeta function SHAI HARAN 93

A prime orbit theorem for self-similar flows and Diophantine approximation MICHEL L. LAPIDUS AND MACHIEL VAN FRANKENHUYSEN 113

The 1022-nd zero of the Riemann zeta function A. M. 0DLYZKO 139

Spectral theory, dynamics, and Selberg's zeta function for Kleinian groups PETER PERRY 145

On zeroes of automorphic £-functions c. SOULE 167

Artin L-functions of graph coverings H. M. STARK AND A. A. TERRAS 181

vii

Preface

These are the proceedings of the Special Session on 'Dynamkal, Spectral and Arithmetic Zeta Functions', held at the Annual Meeting of the American Mathe-matical Society in San Antonio, Texas, January 13-16, 1999.

The purpose of the session was to bring together leading researchers working in these fields, thereby helping to find links, analogies and explore new methods. The volume consists of eleven papers, seven of which were presented at the meet-ing. The subjects of the papers range from dynamical systems or spectral geometry on hyperbolic manifolds (Chang-Mayer, Fan-Jorgenson) and dynamical zeta func-tions (Fel'shtyn, Lapidus-van Frankenhuysen, Stark-Terras), to trace formulas in geometry (Deninger-Singhof, Perry) and in arithmetic (Haran, Soule) as well as to computational work on the Riemann zeta function (Galway, Odlyzko).

We hope that the reader will benefit from the different perspectives of the var-ious authors.

July, 2001 Michel L. Lapidus Machiel van Frankenhuysen

Alphabetical list of contributors with affiliation

Cheng-Hung Chang Institut fiir Theoretische Physik Technische Universitat Clausthal Arnold-Sommerfeld-Straf3e 6 38678 Clausthal-Zellerfeld, Germany ch.chang@tu-clausthal.de

Carol E. Fan Department of Mathematics MC-8130 Loyola Marymount University Los Angeles, CA 90045-8130 cfan@lmumail.lmu.edu

ix

Christopher Deninger Mathematisches Institut WesWilische Wilhelms-Universitat Einsteinstraf3e 62 48149 Munster, Germany deninge@math. uni-muenster .de

Alexander Fel'shtyn Institut fur Mathematik Ernst-Moritz-Arndt-Universitat Jahn-Straf3e 15a D-17 489 Greifswald, Germany felshtyn @mail. uni-greifswald.de

X PREFACE

William F. Galway Department of Mathematics University of Illinois 1409 W. Green Street Urbana, IL 61801 galway@math. uiuc.edu

Jay Jorgenson Department of Mathematics City College of New York New York, NY 10031 jjorgen@mathO.sci.ccny.cuny.edu

Dieter H. Mayer Institut fur Theoretische Physik Technische Universitiit Clausthal Arnold-Sommerfeld-StraBe 6 38678 Clausthal-Zellerfeld, Germany dieter .mayer@tu-clausthal.de

Peter Perry Department of Mathematics University of Kentucky Lexington, Kentucky 40506-0027 perry@ms.uky.edu

Christophe Soule Institut des Hautes Etudes Scientifiques 35, Route de Chartres 91440, Bures-sur-Yvette, France soule@ihes.fr

Audrey A. Terras Department of Mathematics 9500 Gilman Drive University of California, San Diego La Jolla, CA 92093-0112 aterras@ucsd.edu

Shai Haran Department of Mathematics Technion Haifa 32000, Israel haran@tx. technion.ac.il

Michel L. Lapidus Department of Mathematics University of California Riverside, CA 92521-0135 lapidus@math. ucr.edu

Andrew M. Odlyzko AT&T Labs- Research Florham Park, NJ 07932 amo@research.att .com

Wilhelm Singhof Mathematisches Institut Universitiit Dusseldorf UniversitiitsstraBe 1 40225 Dusseldorf, Germany singhof@cs. uni-duesseldorf.de

Harold M. Stark Department of Mathematics 9500 Gilman Drive University of California, San Diego La Jolla, CA 92093-0112 stark@math. ucsd.edu

Machiel van Frankenhuysen Rutgers University Department of Mathematics 110 Frelinghuysen Road Piscataway, NJ 08854-8019 machiel@math.rutgers.edu

Selected Titles in This Series (Continued from the front of this publication)

266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and analytic geometry, 2000

265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique: Homotopy theory in the Swiss Alps, 2000

264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Eflm I. Zelmanov, Editors, Combinatorial and computational algebra, 2000

263 Yan Guo, Editor, Nonlinear wave equations, 2000 262 Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors,

Crystallographic groups and their generalizations, 2000 261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability

on algebraic structures, 2000 260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in

analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000

259 D. V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and its applications, 2000

258 Karsten Grove, Ib Henning Madsen, and Erik Kjaer Pedersen, Editors, Geometry and topology: Aarhus, 2000

257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000

256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000

255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear POE's, dynamics and continuum physics, 2000

254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000

253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000

252 Nathaniel Dean, Cassandra M. McZeal, and Pamela J. Williams, Editors, African Americans in Mathematics II, 1999

251 Eric L. Grinberg, Shiferaw Berhanu, Marvin I. Knopp, Gerardo A. Mendoza, and Eric Todd Quinto, Editors, Analysis, geometry, number theory: The Mathematics of Leon Ehrenpreis, 2000

250 Robert H. Gilman, Editor, Groups, languages and geometry, 1999 249 Myung-Hwan Kim, JohnS. Hsia, Yoshiyuki Kitaoka, and Rainer

Schulze-Pillot, Editors, Integral quadratic forms and lattices, 1999 248 Naihuan Jing and Kailash C. Misra, Editors, Recent developments in quantum

affine algebras and related topics, 1999 247 Lawrence Wasson Baggett and 'David Royal Larson, Editors, The functional and

harmonic analysis of wavelets and frames, 1999 246 Marcy Barge and Krystyna Kuperberg, Editors, Geometry and topology in

dynamics, 1999 245 Michael D. Fried, Editor, Applications of curves over finite fields, 1999 244 Leovigildo Alonso Tarrio, Ana Jeremias L6pez, and Joseph Lipman, Studies in

duality on noetherian formal schemes and non-noetherian ordinary schemes, 1999

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej.

Dynamical, Spectral, and Arithmetic Zeta Functions Michel L. Lapidus and Machiel van Frankenhuysen, Editors

The original zeta function was studied by Riemann as part of his investigation of the distri-bution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the develop-ment of L-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results. This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeti ng of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analo-gies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function. Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a compre-hensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.

ISBN 0- 8218-2079 - 6

9 780821 820797

CONM/290