Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

447 Pages·2003·8.39 MB·English

by  John Derbyshire

P O
RIME BSESSION

P O
RIME BSESSION
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

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
vii
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
PROLOGUE
I
n 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:
ix