http://books.nap.edu/catalog/10532.html PRIME OBSESSION Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html PRIME OBSESSION Bernhard Riemann and the Greatest Unsolved Problem in Mathematics John Derbyshire Joseph Henry Press Washington, D.C. Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html Joseph Henry Press • 500 Fifth Street, NW • Washington, DC 20001 The Joseph Henry Press, an imprint of the National Academies Press, was created with the goal of making books on science, technology, and health more widely available to professionals and the public. Joseph Henry was one of the early founders of the National Academy of Sciences and a leader in early American science. Any opinions, findings, conclusions, or recommendations expressed in this volume are those of the author and do not necessarily reflect the views of the National Academy of Sciences or its affiliated institutions. Library of Congress Cataloging-in-Publication Data Derbyshire, John. Prime obsession : Bernhard Riemann and the greatest unsolved problem in mathematics / John Derbyshire. p. cm. Includes index. ISBN 0-309-08549-7 1. Numbers, Prime. 2. Series. 3. Riemann, Bernhard, 1826-1866. I. Title. QA246.D47 2003 512'.72—dc21 2002156310 Copyright 2003 by John Derbyshire. All rights reserved. Printed in the United States of America. Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html For Rosie Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html CONTENTS Prologue . ix Part I The Prime Number Theorem 1 Card Trick . 3 2 The Soil, the Crop . 19 3 The Prime Number Theorem . 32 4 On the Shoulders of Giants . 48 5 Riemann’s Zeta Function . 63 6 The Great Fusion . 82 7 The Golden Key, and an Improved Prime Number Theorem . 99 8 Not Altogether Unworthy . 118 9 Domain Stretching . 137 10 A Proof and a Turning Point . 151 Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File providedvii by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html viii PRIME OBSESSION Part II The Riemann Hypothesis 11 Nine Zulu Queens Ruled China . 169 12 Hilbert’s Eighth Problem . 184 13 The Argument Ant and the Value Ant . 201 14 In the Grip of an Obsession . 223 15 Big Oh and Möbius Mu . 238 16 Climbing the Critical Line . 252 17 A Little Algebra . 265 18 Number Theory Meets Quantum Mechanics . 280 19 Turning the Golden Key . 296 20 The Riemann Operator and Other Approaches . 312 21 The Error Term . 327 22 Either It’s True, or Else It Isn’t . 350 Epilogue . 362 Notes . 365 Appendix: The Riemann Hypothesis in Song . 393 Picture Credits . 405 Index . 407 Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html PROLOGUE In August 1859, Bernhard Riemann was made a corresponding member of the Berlin Academy, a great honor for a young mathematician (he was 32). As was customary on such occasions, Riemann presented a paper to the Academy giving an account of some research he was engaged in. The title of the paper was: “On the Number of Prime Numbers Less Than a Given Quan- tity.” In it, Riemann investigated a straightforward issue in ordinary arithmetic. To understand the issue, ask: How many prime numbers are there less than 20? The answer is eight: 2, 3, 5, 7, 11, 13, 17, and 19. How many are there less than one thousand? Less than one million? Less than one billion? Is there a general rule or formula for how many that will spare us the trouble of counting them? Riemann tackled the problem with the most sophisticated math- ematics of his time, using tools that even today are taught only in advanced college courses, and inventing for his purposes a math- ematical object of great power and subtlety. One-third of the way into the paper, he made a guess about that object, and then remarked: Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File providedix by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html x PRIME OBSESSION One would, of course, like to have a rigorous proof of this, but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation. That casual, incidental guess lay almost unnoticed for decades. Then, for reasons I have set out to explain in this book, it gradually seized the imaginations of mathematicians, until it attained the sta- tus of an overwhelming obsession. The Riemann Hypothesis, as that guess came to be called, re- mained an obsession all through the twentieth century and remains one today, having resisted every attempt at proof or disproof. Indeed, the obsession is now stronger than ever since other great old open problems have been resolved in recent years: the Four-Color Theo- rem (originated 1852, proved in 1976), Fermat’s Last Theorem (origi- nated probably in 1637, proved in 1994), and many others less well known outside the world of professional mathematics. The Riemann Hypothesis is now the great white whale of mathematical research. The entire twentieth century was bracketed by mathematicians’ preoccupation with the Riemann Hypothesis. Here is David Hilbert, one of the foremost mathematical intellects of his time, addressing the Second International Congress of Mathematicians at Paris in Au- gust 1900: Essential progress in the theory of the distribution of prime num- bers has lately been made by Hadamard, de la Vallée Poussin, von Mangoldt and others. For the complete solution, however, of the problems set us by Riemann’s paper “On the Number of Prime Numbers Less Than a Given Quantity,” it still remains to prove the correctness of an exceedingly important statement of Riemann, viz.... There follows a statement of the Riemann Hypothesis. A hun- dred years later, here is Phillip A. Griffiths, Director of the Institute for Advanced Study in Princeton, and formerly Professor of Math- Copyright © 2003 National Academy of Sciences. All rights reserved. Unless otherwise indicated, all materials in this PDF File provided by the National Academies Press (www.nap.edu) for research purposes are copyrighted by the National Academy of Sciences. Distribution, posting, or copying is strictly prohibited without written permission of the NAP. Generated for [email protected] on Tue Aug 19 11:00:31 2003 http://books.nap.edu/catalog/10532.html PROLOGUE xi ematics at Harvard University. He is writing in the January 2000 issue of American Mathematical Monthly, under the heading: “Research Challenges for the 21st Century”: Despite the tremendous achievements of the 20th century, dozens of outstanding problems still await solution.