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

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) 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. Visit the AMS home page at URL: http: I /vww. ams. org/ 10987654321 06 05 04 03 02 01 Contents

Preface ix

Eigenfunctions of the transfer operators and the period functions for modular groups 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 Christopher Deninger Institut fiir Theoretische Physik Mathematisches Institut Technische Universitat Clausthal WesWilische Wilhelms-Universitat Arnold-Sommerfeld-Straf3e 6 Einsteinstraf3e 62 38678 Clausthal-Zellerfeld, Germany 48149 Munster, Germany [email protected] deninge@math. uni-muenster .de

Carol E. Fan Alexander Fel'shtyn Department of Mathematics MC-8130 Institut fur Mathematik Loyola Marymount University Ernst-Moritz-Arndt-Universitat Los Angeles, CA 90045-8130 Jahn-Straf3e 15a [email protected] D-17 489 Greifswald, Germany felshtyn @mail. uni-greifswald.de

ix X PREFACE

William F. Galway Shai Haran Department of Mathematics Department of Mathematics University of Illinois Technion 1409 W. Green Street Haifa 32000, Israel Urbana, IL 61801 haran@tx. technion.ac.il galway@math. uiuc.edu

Jay Jorgenson Michel L. Lapidus Department of Mathematics Department of Mathematics City College of New York University of California New York, NY 10031 Riverside, CA 92521-0135 [email protected] lapidus@math. ucr.edu

Dieter H. Mayer Andrew M. Odlyzko Institut fur Theoretische Physik AT&T Labs- Research Technische Universitiit Clausthal Florham Park, NJ 07932 Arnold-Sommerfeld-StraBe 6 [email protected] .com 38678 Clausthal-Zellerfeld, Germany dieter [email protected]

Peter Perry Wilhelm Singhof Department of Mathematics Mathematisches Institut University of Kentucky Universitiit Dusseldorf Lexington, Kentucky 40506-0027 UniversitiitsstraBe 1 [email protected] 40225 Dusseldorf, Germany singhof@cs. uni-duesseldorf.de

Christophe Soule Harold M. Stark Institut des Hautes Etudes Scientifiques Department of Mathematics 35, Route de Chartres 9500 Gilman Drive 91440, Bures-sur-Yvette, France University of California, San Diego [email protected] La Jolla, CA 92093-0112 stark@math. ucsd.edu Audrey A. Terras Machiel van Frankenhuysen Department of Mathematics Rutgers University 9500 Gilman Drive Department of Mathematics University of California, San Diego 110 Frelinghuysen Road La Jolla, CA 92093-0112 Piscataway, NJ 08854-8019 [email protected] [email protected] 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