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

Joseph Henry Press500 Fifth Street, NWWashington, DC20001

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.

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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 0-309-14126-5 Mobipocket ISBN

ISBN 0-309-08549-7

1. Numbers, Prime. 2. Series. 3. Riemann, Bernhard, 1826-1866. I. Title.

QA246.D47 2003

512'.72dc21

2002156310

Copyright 2008/2009 Mobipocket.com. All rights reserved.

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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 Quantity. 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 mathematics of his time, using tools that even today are taught only in advanced college courses, and inventing for his purposes a mathematical object of great power and subtlety. One-third of the way into the paper, he made a guess about that object, and then remarked:

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 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 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 time, addressing the Second International Congress of Mathematicians at Paris in August 1900:

Essential progress in the theory of the distribution of prime numbers has lately been made by Hadamard, de la Valle 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 hundred years later, here is Phillip A. Griffiths, Director of the Institute for Advanced Study in Princeton, and formerly Professor of Mathematics 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. 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 twentieth century was the rise of private institutes for mathematical research, funded by wealthy math enthusiasts. Both the Clay Mathematics Institute (founded by Boston financier Landon T. Clay in 1998) and the American Institute of Mathematics (established in 1994 by California entrepreneur John Fry) have targeted the Riemann Hypothesis. The Clay Institute has offered a prize of one million dollars for a proof or a 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 can easily grasp. It lies deep in the heart of some quite abstruse mathematical theory. Here it is:

The Riemann Hypothesis

All non-trivial zeros of the zeta function have real part one-half.

To an ordinary reader, even a well-educated one, who has had no advanced mathematical training, this is probably quite incomprehensible. It might as well be written in Old Church Slavonic. In this book, as well as 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 as being too cute) contain mathematical exposition, leading the reader, gently I hope, to an understanding of the Riemann Hypothesis and its importance. The even-numbered chapters offer historical and biographical background matter.

I originally intended these two threads to be independent, so that readers who don't like equations and formulae could read only the even-numbered chapters while readers who did not care for history or anecdote could just read the odd-numbered ones. I did not quite manage to hold to this plan all the way through, and I now doubt that it can be done with a subject so intricate. Still, the basic pattern was not altogether lost. There is much more math in the odd-numbered chapters, and much less in the even-numbered ones, and you are, of course, free to try reading just the one group or the other. I hope, though, that you will read the whole book.