Surveys in Number Theory: Papers from the Millennial Conference on Number Theory PDF
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Surveys in Number Theory Surveys in Number Theory Papers from the Millennial Conference on Number Theory Edited by M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand, and W Philipp Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business AN A K PETERS BOOK CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 First issued in hardback 2019 © 2003 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Govermnent works ISBN-l3: 978-1-56881-162-8 (pbk) ISBN-13: 978-1-138-45428-6 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Library of Congress Cataloging-in-Publication Data Millennial Conference on Number theory (2000 : University of Illinios at Urbana-Champaign) Surveys in Number theory; papers from the Millennial Conference on Number Theory / edited by M. A. Bennett ... let al.). p.cm. Conference held at the University of Illinois at Urbana-Champaign, May 21-26, 2000. Includes bibliographical references. ISBN 1-56881-162-4 1. Number theory-Congresses. I. Bennett, M. A. (Michael A.), 1966-II. Title. QA241 .M5342000 512' .7-dc21 2002070231 Contents Preface xi Completeness Problems and the Riemann Hypothesis: An Annotated Bibliography 1 Michel Balazard A Survey on Pure and Mixed Exponential Sums Modulo Prime Powers 29 Todd Cochrane and Zhiyong Zheng One Hundred Years of Normal Numbers 57 Glyn Harman On Theorems of Barban-Davenport-Halberstam Type 75 C. Hooley Integer Points, Exponential Sums and the Riemann Zeta Function 109 Μ. N. Huxley Recent Developments in Automorphic Forms and Applications 125 Wen-Ching Winnie Li Convergence of Corresponding Continued Fractions 149 Lisa Lorentzen On the Analytic Continuation of Various Multiple Zeta-Functions 169 Kohji Matsumoto Quelques Remarques sur la Theorie d’lwasawa des Courbes Elliptiques 193 Bernadette Perrin-Riou Computing Rational Points on Curves 223 Bjorn Poonen G. H. Hardy As I Knew Him 247 Robert A. Rankin Some Applications of Diophantine Approximation 261 R. Tijdeman Waring’s Problem: A Survey 285 R. C. Vaughan and T. D. Wooley Solving the Pell Equation 325 H C. Williams v Preface The Millennial Conference on Number Theory was held on May 21-26, 2000 on the campus of the University of Illinois at Urbana-Champaign. The meeting was organized by M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, and A. J. Hildebrand of the Mathematics Department and W. Philipp of the Statistics Department at the University of Illinois. A total of 276 mathematicians from 30 countries were present at the meeting. The proceedings of this conference, containing 72 papers based on lec tures given at the conference, have been published separately in three vol umes under the title Number Theory for the Millennium. The present volume contains fourteen of these papers which represent broad surveys of topics in number theory or related areas. We hope that these papers will help stimulate interest in these areas and will be of lasting value as references on the topics covered. Urbana, Illinois 7 December 2001 M. A. Bennett B. C. Berndt N. Boston H. G. Diamond A. J. Hildebrand W. Philipp Vll Completeness Problems and the Riemann Hypothesis: An Annotated Bibliography Michel Balazard 1 Notation and Basic Formulas • We consider complex functions / defined and measurable on the posi tive half-line (0, +00). Functions defined on some subset E of (0, +00) are extended to (0, +00) by assigning the value 0 outside E. • For λ > 0, the dilation operator K\ is defined by • The letter s denotes a complex variable, σ its real part and r its imaginary part. • The Mellin transform (1.1) is defined for every s such that the integral converges absolutely; it is holomorphic in the region where it is defined. • For / in L2(0, +00), Mf(s) is a priori defined nowhere, but according to the Plancherel theory, it can be defined a.e. on the line σ = 1/2 by setting The operator / (2π) l/2M f is a unitary operator from L2(0, +00) to L2(l/2-HK) =: L2. • The letter p denotes two different objects: the “fractional part func tion” p(x) := x — Ix\, and a generic complex number p = β + «7, β G M, 7 G R, usually a zero of some Mellin transform. No confusion should occur from this usage. 1
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