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Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics John Derbyshire PRIME OBSESSION PRIME OBSESSION Bernhard Riemann and the Greatest Unsolved Problem in Mathematics John Derbyshire Joseph Henry Press Washington, D.C. 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. For Rosie vii 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 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 ix 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: [...]... “Introduction to the Analysis of the Infinite.” The notions of the infinite and the infinitesimal created serious problems in math during the early nineteenth century, though, and eventually they were swept away altogether in a great reform Modern analysis does not admit these concepts They linger on in the vocabulary of mathematics, and I shall make free use of the word “infinity” in this book This usage,... mathematicians are more interested in convergent series than divergent ones V Suppose that instead of moving one inch to the right, then a half-inch to the right, then a quarter-inch to the right, and so on, I decided to alternate directions: an inch to the right, a half-inch to the left, a quarter-inch to the right, an eighth-inch to the left.… After seven moves I’d be at the point shown in Figure 1-1 0 CARD TRICK... pencil point is now on the one-inch mark and you have moved it a total of one inch (see Figure 1-7 ) 64ths 1 2 FIGURE 1-7 3 CARD TRICK 11 Now move the pencil half an inch further to the right (see Figure 1-8 ) 64ths 1 2 3 FIGURE 1-8 Now move the pencil a quarter-inch further to the right … then an eighth-inch … then a sixteenth … then a thirty-second … then a sixty-fourth Your pencil is in the position... the tremendous achievements of the 20th century, dozens of outstanding problems still await solution Most of us would probably agree that the following three problems are among the most challenging and interesting The Riemann Hypothesis The first is the Riemann Hypothesis, which has tantalized mathematicians for 150 years An interesting development in the United States during the last years of the. .. these infinite series dwell in is what mathematicians call analysis Analysis used, in fact, to be thought of as the study of the infinite, that is, the infinitely large, and of the infinitesimal, the infinitely small When Leonhard Euler—of whom I shall write much more later—published the first great textbook of analysis in 1748, he called it Introductio in analysin infinitorum: “Introduction to the Analysis... equation, or the whole fascinating issue—just recently come to life after long dormancy—of the moments of the zeta function Nor is there any mention of the Generalized Riemann Hypothesis, the Modified Generalized Riemann Hypothesis, the Extended Riemann Hypothesis, the Grand Riemann Hypothesis, the Modified Grand Riemann Hypothesis, or the Quasi -Riemann Hypothesis Even more distressing, there are many... disproof; the American Institute of Mathematics has addressed the Hypothesis with three full-scale conferences (1996, 1998, and 2002), attended by researchers from all over the world Whether these new approaches and incentives will crack the Riemann Hypothesis at last remains to be seen Unlike the Four-Color Theorem, or Fermat’s Last Theorem, the Riemann Hypothesis is not easy to state in terms a nonmathematician... describing the history of the Hypothesis, and some of the personalities who have been involved with it, I have attempted to bring this deep and mysterious result within the understanding of a general readership, giving just as much mathematics as is needed to understand it ***** The plan of the book is very simple The odd-numbered chapters (I was going to make it the prime- numbered, but there is such a thing... 1976), Fermat’s Last Theorem (originated 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... attained the status of an overwhelming obsession The Riemann Hypothesis, as that guess came to be called, remained 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 Theorem (originated 1852, proved in . immediately. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics John Derbyshire PRIME OBSESSION PRIME OBSESSION Bernhard Riemann and the Greatest Unsolved Problem in Mathematics John. affiliated institutions. Library of Congress Cataloging -in- Publication Data Derbyshire, John. Prime obsession : Bernhard Riemann and the greatest unsolved problem in mathematics / John Derbyshire. . going to make it the prime- numbered, but there is such a thing as being too cute) contain mathematical exposition, leading the reader, gently I hope, to an understanding of the Riemann Hypoth- esis