PRIME OBSESSION

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 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:

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:

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 Math-

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”:

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 incomprehen-

sible. 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.

I have aimed this book at the intelligent and curious but nonmathematical reader. That statement, of course, raises a number of questions. What do I mean by “nonmathematical?” How much math knowledge have I assumed my readers possess? Well, everybody knows some math. Probably most educated people have at least an inkling of what calculus is all about. I think I have pitched my book to the level of a person who finished high school math satisfactorily and perhaps went on to a couple of college courses. My original goal was, in fact, to explain the Riemann Hypothesis without using any calculus

at all. This proved to be a tad over-optimistic, and there is a very small quantity of very elementary calculus in just three chapters, explained as it goes along.

Pretty much everything else is just arithmetic and basic algebra: multiplying out parentheses like (a + b) × (c + d), or rearranging equations so that S = 1 + xS becomes S = 1/(1–x). You will also need a willingness to take in the odd shorthand symbols mathematicians use to spare the muscles of their writing hands. I claim at least this much: I don’t believe the Riemann Hypothesis can be explained using math more elementary than I have used here, so if you don’t understand the Hypothesis after finishing my book, you can be pretty sure you will never understand it.

* * * * *

Various professional mathematicians and historians of mathematics were generous with their help when I approached them. I am profoundly grateful to the following for their time, freely given, for their advice, sometimes not taken, for their patience in dealing with my repetitive dumb questions, and in one case for the hospitality of his home: Jerry Alexanderson, Tom Apostol, Matt Brin, Brian Conrey, Harold Edwards, Dennis Hejhal, Arthur Jaffe, Patricio Lebeuf, Stephen Miller, Hugh Montgomery, Erwin Neuenschwander, Andrew Odlyzko, Samuel Patterson, Peter Sarnak, Manfred Schröder, Ulrike Vorhauer, Matti Vuorinen, and Mike Westmoreland. Any gross errors in this book’s math are mine, not theirs. Brigitte Brüggemann and Herbert Eiteneier helped plug the gaps in my German. Commissions from my friends at National Review, The New Criterion, and The Washington Times allowed me to feed my children while working on this book. Numerous readers of my online opinion columns helped me understand what mathematical ideas give the most difficulty to nonmathematicians.

Along with these acknowledgments goes an approximately equal number of apologies. The topic this book deals with has been under

intensive investigation by some of the best minds on our planet for a hundred years. In the space available to me, and by the methods of exposition I have decided on, it has proved necessary to omit entire large regions of inquiry relevant to the Riemann Hypothesis. You will find not one word here about the Density Hypothesis, the approximate functional 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 workers who have toiled away valiantly in these vineyards for decades, but whose names are absent from my text: Enrico Bombieri, Amit Ghosh, Steve Gonek, Henryk Iwaniec (half of whose mail comes to him addressed as “Henry K. Iwaniec”), Nina Snaith, and many others. My sincere apologies. I did not realize, when starting out, what a vast subject I was taking on. This book could easily have been three times, or thirty times, longer, but my editor was already reaching for his chainsaw.

And one more acknowledgment. I hold the superstitious belief that any book above the level of hired drudge work—any book written with care and affection—has a presiding spirit. By that, I only mean to say that a book is about some one particular human being, who is in the author’s mind while he works, and whose personality colors the book. (In the case of fiction, I am afraid that all too often that human being is the author himself.)

The presiding spirit of this book, who seemed often to be glancing over my shoulder as I wrote, whom I sometimes imagined I heard clearing his throat shyly in an adjoining room, or moving around discreetly behind the scenes in both my mathematical and historical chapters, has been Bernhard Riemann. Reading him, and reading about him, I developed an odd mixture of feelings for the man: great sympathy for his social awkwardness, wretched health, repeated be-

reavements, and chronic poverty, mixed with awe at the extraordinary powers of his mind and heart.

A book should be dedicated to someone living, so that the dedication can give pleasure. I have dedicated this book to my wife, who knows very well how sincere that dedication is. There is a sense, though, not to be left unremarked in a prologue, in which this book most properly belongs to Bernhard Riemann, who, in a short life blighted with much misfortune, gave to his fellow men so very, very much of everlasting value—including a problem that continues to vex them a century and a half after, in a characteristically diffident aside, he noted his own “fleeting vain attempts” to resolve it.

John Derbyshire

Huntington, New York

June 2002