Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)

I. There is a satisfying symmetry about the fact that the Riemann Hypothesis (RH), after 120 years among the mathematicians, has got the attention of the physicists. Riemann’s own imagination was, as I noted in Chapter 10.i, very much that of a physical scientist. “Four of the nine papers that he himself managed to publish must be viewed as belonging to physics” (Laugwitz). And in fact, number theorist Ulrike Vorhauer¹³² reminds me, the distinction between mathematician and physicist was not much made in Riemann’s time. Shortly before that it was not made at all. Gauss was a first-rank physicist as well as a first-rank mathematician and would have been puzzled to hear the two disciplines spoken of as separate spheres of interest.

Jonathan Keating¹³³ tells the following anecdote, which I must say I find rather eerie.

Of course, Jonathan adds, Riemann didn’t have twentieth-century operator algebra to help him with the perturbation problem, to give him the set of all possible wobble frequencies as a spectrum of eigenvalues. He’d just slogged through the differential equations, creating a sort of ad hoc, embryonic operator theory for himself. Still, it’s hard to believe that a mind as acute and penetrating as Riemann’s would have missed the analogy between the zeta zeros strung out on the critical line, and his spectrum of perturbation frequencies—the analogy that was so dramatically paralleled over afternoon tea in Fuld Hall 113 years later!

II. It was at New York University’s Courant Institute that I heard Keating tell that anecdote, in the early summer of 2002. The occasion was a four-day series of lectures and discussions organized by the American Institute of Mathematics (AIM). The title of the thing was “Workshop on Zeta Functions and Associated Riemann Hypotheses.”

There were many famous names at the Courant conference. Atle Selberg himself showed up, 84 years old and still sharp as a tack. (He pulled up Peter Sarnak on a point of historical-mathematical fact in the very first lecture. During lunch break I went up to the Courant’s excellent library and checked the point. Selberg was right.) Many of the other names mentioned in these last few chapters were present, too, including both halves of the Montgomery-Odlyzko Law. Other attendees included the current superstar of math, Andrew Wiles, famous for having proved Fermat’s Last Theorem; Harold Edwards, whose definitive book on the zeta function I have mentioned several times in these pages; and Daniel Bump, one of the two names attached to the most euphonious of all RH-related results, the Bump-Ng Theorem.¹³⁴

The AIM has been a considerable force in assaults on the RH during recent years. The Courant conference was the third they had sponsored on RH-related topics. The first, at the University of Washington in Seattle, in August 1996, was inspired by a wish to commemorate the proof of the Prime Number Theorem by Hadamard and de la Vallée Poussin 100 years earlier. The second was held in 1998 at the Erwin Schrödinger Institute in Vienna. The AIM by no means restricts its activities to RH studies—nor even just to number theory. They currently have a project on general relativity, for example. They have, though, done great work in bringing together scholars from different fields, pursuing all the different approaches I have mentioned: algebraic, analytic, computational, and physical.

AIM was established in 1994 by Gerald Alexanderson, a senior figure in American mathematics (and author of a very good book about George Pólya), and John Fry, a California businessman. Fry comes from a family of entrepreneurs. His parents owned a successful chain of supermarkets in California. John fell in love with math early on and in the 1970s he majored in the subject at Santa Clara University, where Alexanderson was on the faculty. After graduation John faced the choice of following the family tradition into business or going to graduate school. John opted for business and with his two

brothers started the Fry’s Electronics chain of stores, originally just in California, but at the time of writing going nationwide.

John Fry and Jerry Alexanderson stayed in touch. They shared a common interest, collecting rare math books and original papers. In the early 1990s they kicked around the idea of establishing a math library to house their collections. This developed into a plan for a math institute. They called in Brian Conrey, an old classmate of John’s at Santa Clara, a number theorist of some repute, and a very successful head of department at Oklahoma State University.

For the first few years of its existence, AIM was funded almost entirely by personal donations from John Fry, to the tune of around $300,000 a year. This was a case of doing good by stealth. John is a reserved and private man who does not publicize his activities. When I first learned about AIM I went looking for a picture of him on the internet; there weren’t any. In his element, though, that is, among mathematicians and people who love math, John is perfectly accessible. He took a party of us to lunch at the Courant conference in New York. A tall, boyish man, his face lights up when he talks math. I quietly wondered whether he had ever regretted the decision to go into business rather than the academy, but thought it might be impertinent to ask, and so missed the opportunity.

Visiting AIM headquarters a few days before the Courant conference, I found it occupying a utilitarian suite of rooms attached to the Fry’s store in Palo Alto, California. In 2001, however, AIM applied for National Science Foundation funding to help establish a conference center on a leafy 200-acre property south of San Jose, California. The funding was approved, and research programs at the new location will begin in December 2002.

Another privately-funded enterprise similar to AIM began on the East Coast of the United States in 1998, when Boston businessman Landon T. Clay and Harvard mathematician Arthur Jaffe established the Clay Mathematics Institute (CMI). While AIM’s first major initiative was to commemorate the proof of the Prime Number Theo-

rem, CMI’s was to mark the anniversary of Hilbert’s speech at the 1900 Paris Congress.

For that purpose, the Clays held a two-day Millennium Event, also in Paris, at the Collège de France, in May 2000, during the course of which a $7 million fund was unveiled, $1 million to be awarded for the solution to each of seven great mathematical problems. The RH was naturally included, as problem number 4. (The order was based on the lengths of the problems’ titles, to give the announcement an attractive appearance.)Whatever may be the case with the other six problems, $1 million is very little extra incentive to prove, or disprove, the Hypothesis. It is sufficiently established as the open problem in math at the beginning of the twenty-first century that whoever can resolve it will attain, in addition to everlasting fame, financial success—in lecture, interview, and royalty fees alone—far in excess of $1 million.¹³⁵

III. What are the prospects for a proof or disproof of the RH? Delivering prognostications about this sort of thing is a very good way to make a fool of yourself. This is true even if you are a great mathematician, which of course I am not. Seventy-five years ago, lecturing to a lay audience, David Hilbert ranked three problems in ascending order of difficulty:

• The RH.

• Fermat’s Last Theorem.

• “The Seventh”—that is, number 7 in the list of 23 problems Hilbert presented at the 1900 congress. In its more explicit form: If a and b are algebraic numbers, then a^b is transcendental (see Chapter 11.ii) except when it trivially isn’t.

Hilbert said that the RH would be resolved in his lifetime, and Fermat’s Last Theorem within the lifetime of younger audience mem-

bers; but “no-one in this room will live to see a proof of the Seventh.” In fact the Seventh was proved less than 10 years later, by Alexander Gel’fond and Theodor Schneider working independently. Hilbert was right, at a stretch, about Fermat’s Last Theorem, proved by Andrew Wiles in 1994, when younger members of Hilbert’s audience would have been in their nineties. He was drastically wrong about the RH, though. Should the RH make a fool of me, too—should the words I am about to write be rendered null and void by a proof of the RH turning up while this book is at the casebinder—I shall at least be able to console myself that I am in excellent company.

I am, therefore, going to stick my neck out and say that I believe a proof of the RH to be a long way beyond our present grasp. Surveying the modern history of attempts on the RH is something like reading an account of a long and difficult war. There are sudden surprising advances, tremendous battles, and heartbreaking reverses. There are lulls—times of exhaustion, when each side, “fought out,” does little but conduct small-unit probes of the enemy defenses. There are breakthroughs followed by outbursts of enthusiasm; and there are stalemates followed by spells of apathy.

My impression of the current (mid-2002) state of affairs— though, to be sure, it is only the impression of a noncombatant—is that researchers are stalemated. We are in a lull. The great burst of interest generated by Deligne’s proof of the Weil Conjectures in 1973 and by the Montgomery-Odlyzko developments of 1972–1987 seems to me to have spent itself.

In May 2002 I spent three days at the AIM office in Palo Alto, reviewing the videotaped record of the 1996 Seattle conference. The following month I attended the Courant Institute workshop. If you subtract 1996 from 2002, you get six years. If you “subtract” the contents of the Seattle conference from those of the Courant workshop, the mathematicians assembled at the Courant had little new to show. That is not a very surprising statement, to be sure, and I certainly do not mean it in a disparaging sense. This is work of the utmost difficulty. Progress is naturally slow, and six years is a very short time in

the history of mathematics. (It took 357 years to prove Fermat’s Last Theorem!) And there were some striking presentations at the Courant by younger mathematicians like Ivan Fesenko.

Still, the overriding impression was of stalemate. It is as if the RH were a mountain to be climbed, but from whichever direction one approaches it, one sooner or later finds oneself stuck at the rim of a wide, bottomless crevasse. I lost count of the number of times, in both 1996 and 2002, a lecturer ended his presentation with a verbal throwing up of hands: “This is of course a very important advance. However, it is not clear how we can proceed from here to a proof of the classical RH….”

Sir Michael Berry, who has a way with words, has coined the concept of the “clariton,” which he defines to be “the elementary particle of sudden understanding.” In the realm of the RH, claritons are currently in short supply.

Andrew Odlyzko: “It was said that whoever proved the Prime Number Theorem would attain immortality. Sure enough, both Hadamard and de la Vallée Poussin lived into their late nineties. It may be that there is a corollary here. It may be that the RH is false; but, should anyone manage to actually prove its falsehood—to find a zero off the critical line—he will be struck dead on the spot, and his result will never become known.”

IV. Setting aside the search for a proof, how do mathematicians feel about the RH? What does their intuition tell them? Is the RH true, or not? What do they think? I made a point of asking every mathematician I spoke with, very directly, whether he or she believed the Hypothesis to be true. The answers formed a wide spectrum, with a full range of eigenvalues.

Among that majority of mathematicians who believe it true (Hugh Montgomery, for example), it is the sheer weight of evidence that tells. Now, all professional mathematicians are aware that weight

of evidence can be a very treacherous measure. There was a good weight of evidence for Li(x) being always greater than π (x), until Littlewood’s 1914 result disproved it. Ah, yes, RH believers will tell you, but that was merely one line of evidence, numerical evidence, together with the unsupported assumption that the second log-integral term would continue to dominate the difference, which would therefore always be negative. For the Hypothesis we have far more lines. The RH underpins an enormous body of results, most of them very reasonable and—to bring in a word mathematicians are especially fond of—“elegant.” There are now hundreds of theorems that begin, “Assuming the truth of the Riemann Hypothesis….” They would all come crashing down if the RH were false. That is undesirable, of course, so the believers might be accused of wishful thinking, but it’s not the undesirability of losing those results, it’s the fact of their existence. Weight of evidence.

Other mathematicians believe, as Alan Turing did, that the RH is probably false. Martin Huxley¹³⁶ is a current nonbeliever. He justifies his nonbelief on entirely intuitive grounds, citing an argument first put forward by Littlewood: “A long-open conjecture in analysis generally turns out to be false. A long-open conjecture in algebra generally turns out to be true.”

The answer I liked best was Andrew Odlyzko’s. He was actually the first person to whom I posed the question—the first mathematician I approached, when I was preparing the proposal for this book. We went for dinner at a restaurant in Summit, New Jersey. Andrew was at that time working for Bell Labs; he is now at the University of Minnesota.

I was fairly new to the RH at this point and had been learning a lot. With an excellent Italian meal under our belts and two hours of solid math talk behind us, having finally run out of things to ask, I said this:

JD: Andrew, you have gazed on more non-trivial zeros of the Riemann zeta function than any person alive. What do you think about this darn Hypothesis? Is it true, or not?

AO: Either it’s true, or else it isn’t.

JD: Oh, come on, Andrew. You must have some feeling for an answer. Give me a probability. Eighty percent it’s true, twenty percent it’s false? Or what?

AO: Either it’s true, or else it isn’t.

I could get no more from him than that. He simply would not commit himself. In a later conversation, in another place, I asked Andrew if there are any good mathematical reasons to believe the Hypothesis false. Yes, he said, there are some. You can, for example, decompose the zeta function into different parts, each of which tells you something different about zeta’s behavior. One of these parts is the so-called S function. (This has no connection at all with the function I called S(x) in Chapter 9.ii.) For the entire range for which zeta has so far been studied—which is to say, for arguments on the critical line up to a height of around 10²³—S mainly hovers between –1 and +1. The largest value known is around 3.2. There are strong reasons to think that if S were ever to get up to around 100, then the RH might be in trouble. The operative word there is “might”; S attaining a value near 100 is a necessary condition for the RH to be in trouble, but not a sufficient one.

Could values of the S function ever get that big? Why, yes. As a matter of fact, Atle Selberg proved in 1946 that S is unbounded; that is to say, it will eventually, if you go high enough up the critical line, exceed any number you name! The rate of growth of S is so creepingly slow that the heights involved are beyond imagining; but certainly S will eventually get up to 100. Just how far would we have to explore up the critical line for S to be that big? Andrew: “Probably around T equals .” Way beyond the range of our current computational abilities, then? “Oh, yes. Way beyond.”

V. A thing that nonmathematical readers want to know, a question that is always asked when mathematicians address lay audiences, is, What use is it? Suppose the RH were proved true, or false. What prac-

tical consequences would follow? Would our health, our convenience, our safety be improved? Would new devices be invented? Would we travel faster? Have more devastating weapons? Colonize Mars?

I had better unmask myself at this point as a pure mathematician sans mélange, having no interest in such questions at all. Most mathematicians—and most theoretical physicists, too—are motivated not by any thought of advancing the health or convenience of the human race, but by the sheer joy of discovery and the challenge of tackling difficult problems. Mathematicians are generally pleased when their work turns out to have some practical result (at any rate if the result is peaceful), but they rarely think about such things in their working lives. At the Courant conference I sat through four days of solid lectures and discussions on topics related to the RH, from 9:30 A.M. to 6:00 P.M. every day, without ever hearing a mathematician mention practical consequences.

Here is what Jacques Hadamard had to say on this point in The Psychology of Invention in the Mathematical Field.

G.H. Hardy, in the concluding pages of his strange little Apology, was more blunt and more personal about it.

In the case of prime number theory, Hadamard’s “the answer appears to us before the question” applies, and Hardy’s claim is no longer true. Beginning in the late 1970s, prime numbers began to attain great importance in the design of encryption methods for both military and civilian use. Ways to test a large number for primality, ways to resolve large numbers into their prime factors, ways to manufacture gigantic primes; these all became very practical matters indeed in the last two decades of the twentieth century. Theoretical results, including some of Hardy’s, were essential in these developments, which, among other things, allow you to use your credit card to order goods over the internet. A resolution of the RH would undoubtedly have further consequences in this field, validating all those countless theorems about primes that begin, “Assuming the truth of the RH…” and acting as a spur to further discoveries.

And of course, if the physicists really do succeed in identifying a “Riemann dynamics,” our understanding of the physical world will be transformed thereby.

Unfortunately, it is impossible to predict what things will follow from that transformation. Not even the cleverest people can make such predictions, and those who do should not be trusted. Here is a mathematician at work, not quite 100 years ago.

That is from Bertrand Russell’s autobiography. What was stumping him was the attempt to find a definition of “number” in terms of pure logic. What does “three,” for example, actually mean? The German logician Gottlob Frege had come up with an answer; but Russell

had found a flaw in Frege’s reasoning and was searching for a way to plug the leak.

If you had asked Russell, during those summers of frustration, whether his perplexities were likely to lead to any practical application, he would have hooted with laughter. This was the purest of pure intellection, to the degree that even Russell, a pure mathematician by training, found himself wondering what the point was. “It seemed unworthy of a grown man to spend his time on such trivialities…,” he remarked. In fact, Russell’s work eventually brought forth Principia Mathematica, a key development in the modern study of the foundations of mathematics. Among the fruits of that study have been, so far, victory in World War II (or at any rate, victory at a lower cost than would otherwise have been possible) and machines like the one on which I am writing this book.¹³⁷

The RH should therefore be approached in the spirit of Hadamard and Hardy, though preferably without the overlay of melancholy Hardy put on his disclaimer. As Andrew Odlyzko told me, “Either it is true, or else it isn’t.” One day we shall know. I have no idea what the consequences will be, and I don’t believe anyone else has, either. I am certain, though, that they will be tremendous. At the end of the hunt, our understanding will be transformed. Until then, the joy and fascination is in the hunt itself, and—for those of us not equipped to ride—in observing the energy, resolution, and ingenuity of the hunters. Wir müssen wissen, wir werden wissen.