It’s been called the hardest problem in mathematics, and also the most important. Three books have recently been published about it and dozens of the world’s most brilliant mathematicians are devoting themselves to it. A million-
dollar prize awaits the person who solves it. Known as the Riemann Hypothesis, no mathematical problem inspires
such fear and awe — it is said that some mathematicians would sell their souls for the answer.
A decade ago, mathematics splashed onto front pages the world over when Andrew Wiles announced that he had solved another famous problem known as Fermat’s Last Theorem. The public interest in Wiles’ solution stunned the mathematics community — a member of their notoriously nerdy fraternity was now being asked to pose for Gap ads. Suddenly, “It felt almost sexy to be a mathematician,” writes Oxford don Marcus du Sautoy in his new book, The Music of the Primes (HarperCollins). And next month’s publication of David Foster Wallace’s Everything and More, a 300-page rumination on transfinite numbers, seems set to propel math into the lexicon, and onto coffee tables, of the literary cool set.
With a pedigree linking many of the greatest names in the field, the Riemann Hypothesis runs like a river through vast swaths of seemingly distinct mathematical territory. Andrew Wiles himself has compared a proof of this proposition to what it meant for the 18th century when a solution to the longitude problem was found. With longitude licked, explorers could navigate freely around the physical world; so too, if Riemann is resolved, mathematicians will be able to navigate more fluidly across their domain. Its import extends into areas as diverse as number theory, geometry, logic, probability theory and even quantum physics.
The Riemann Hypothesis is a proposal about prime numbers,
the atomic elements of the number system. Primeness is one of the most essential concepts in mathematics, for primes — 2, 3, 5, 7, 11 and so on — are numbers that cannot be broken into any smaller elements. All other integers can be built up by multiplication of these basic units. So, for example, 6 is built up from 2 x 3, 15 from 3 x 5, 49 from 7 x 7. In his book The Riemann Hypothesis (FSG), science writer Karl Sabbagh makes an analogy between numbers and molecules. All of the vast plethora of molecules that inhabit our world,
everything from salt and ammonia to hemoglobin, are
made up of the basic elements of the periodic table — carbon, hydrogen, oxygen and so on. As Sabbagh notes, the primes may be seen as the periodic table of the number system. Yet where the elements follow a clear pattern, the primes seem to be distributed randomly.
To mathematicians, randomness is anathema. As du Sautoy writes, they “can’t bear to admit that there might not be an explanation for the way nature has picked the primes.” That would be like “listening to white noise”; what mathematicians crave above all else is harmony. They want, they need, they demand a pattern behind the apparent chaos. Du Sautoy quotes the great French mathematician and physicist Henri Poincare: “The scientist does not study nature because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing and if nature were not worth knowing, life would not be worth living.”
For many mathematicians life would not be worth living if the organization of the primes did not ultimately conform to some beautiful underlying order. The Riemann Hypothesis proposes what that order might be. One of the most innovative mathematical thinkers of all time, Bernhard Riemann was a sickly German genius of the mid–19th century who also developed the non-Euclidean geometry on which Einstein based his general theory of relativity. Riemann’s life and work form the subject of John Derbyshire’s touching biography Prime Obsession (Joseph Henry Press). Though he died at 39 and his collected works amount to a single slim volume, virtually every paper Riemann wrote revolutionized a different branch of mathematics.
Riemann’s Hypothesis is not easy to state in any language. In essence it links the distribution of prime numbers to a complicated equation called the Riemann Zeta Function. For some values this equation equals zero, and it turns out there are an infinite number of such values, which mathematicians refer to as the “zeros” of the zeta function. Riemann demonstrated that there is a beautiful and unexpected link between these “zeros” and the pattern of the prime numbers.
Each Riemann zero can be represented as a point on something called the complex plane, one of mathematics’ most truly enchanted places. Formed from the intersection
of the “real” and the “imaginary” numbers, the complex plane is also where the fabled Mandelbrot Set lives. To his astonishment, Riemann discovered that on this plane the zeta-function zeros seemed to lie in a strict vertical line, which is now called the critical line. Why this might be so is one of the deepest questions in mathematics. It was Riemann’s intuition, his hypothesis, that all the zeta zeros must lie on this line.
Riemann did not prove his proposition, however, and neither could the finest mathematicians since. John Nash, the hero of A Beautiful Mind, was known to be working on the problem just before he started receiving messages from aliens. Alan Turing, one of the founders of modern computing, attempted to build a cog-and-gear mechanical device for calculating zeta zeros. One prominent Riemannologist was promised a Ferrari by his father if he ever managed to solve the question. Over the years, a number of people have claimed to have proofs, but none have withstood scrutiny.
In 2000, the Riemann Hypothesis was selected by the Clay Mathematics Institute in Boston as one of the seven “Millennium Problems,” each of which is now attached to a million-dollar prize. But money may not be enough — some mathematicians suspect the problem may simply be beyond human capability. Or worse still, that it is unprovable. Another option would be to disprove it, for the discovery of a single zeta zero not on the critical line would immediately render the proposition false. On the surface this would appear an easier row to hoe — all one has to do is to start at the first zero and keep on calculating. Sooner or later, if exceptions exist they must reveal themselves. Though mathematicians would still want to understand why the hypothesis wasn’t true, at least the basic question would be laid to rest.
The idea that there might be exceptions to Riemann’s rule has inspired one
of the more quixotic enterprises in the history of mathematics, a project that
might also be ‰ included among the great conceptual works of our time. For the
past 25 years, Andrew Odlyzko has been calculating Riemann zeta zeros — to date
he has recorded some 30 billion of them! If you go to his Web
site, you can download a list of the first 100,000 Riemann zeros accurate
to nine decimal places, and another of the first 100 zeros accurate to 1,000
decimal places. During his calculations, Odlyzko estimates he has used 250,000
hours of computer time, and has generated 640 gigabytes of data. This vast haul
of unadulterated zeta zeros constitutes one of the largest caches of pure mathematical
information ever assembled.
Originally from Poland, Odlyzko still speaks with a strong but gentle East European accent after 40 years in the U.S. Bespectacled and balding, he looks every inch the classic nerd. As a mathematician Odlyzko has had a lifelong interest in computation, and his research interests have ranged across a wide variety of fields from cryptography and error-correcting codes to number theory, combinatorics, communications networks and electronic publishing. More recently he has become an expert on the economics of e-commerce. Until 2001, Odlyzko was a leading researcher at AT&T’s Bell Labs in New Jersey. Two years ago he moved to the University of Minnesota to take up an appointment in the mathematics department and to head their new Digital Technology Center. Finding zeta zeros is what he does in his spare time.
Speaking by phone from Minneapolis, Odlyzko seems a man of Zen-like patience. His search for Riemann zeros dates back to 1978, when Bell Labs became the first private company to acquire a CRAY-1 supercomputer. Most of the machine’s processing power was devoted to company research, but Bell offered small, five-hour chunks for worthy scientific hobby projects. Odlyzko put in a proposal to calculate the first million zeta zeros. “In the end,” he notes, with typical understatement, “I used up quite a bit more time than that.”
When considering Riemann’s critical line, it is helpful to imagine a narrow cliff face, an infinitely high but very thin vertical wall stretching straight up into the stratosphere. The question can then be stated thus: Is there any place where this vertiginous face deviates from its seeming perfection? Seen in this light, Odlyzko might be viewed as a kind of mathematical rock climber, a minuscule speck inching his way ever higher up an endless, flat escarpment. As he climbs, the numbers defining each zeta zero get progressively larger and his calculations ever more lengthy. The higher he goes, the more computational power he needs.
Odlyzko could have kept on calculating zeta zeros from the bottom up forever.
There is indeed an open-source computer project called ZetaGrid in which volunteers
around the world donate spare cycles on their PCs to check that every Riemann
zero really does lie on the critical line. (See www.zetagrid.net.)
Since August 2001, when the project was initiated, participants have checked
the first 470 billion, and they are currently progressing at around a billion
zeros a day. But Odlyzko had a hunch that if exceptions exist, they would only
be found at very high altitudes. He realized that if he was ever going to find
a counterexample, he’d have to give up the idea of traversing the entire critical
line and restrict himself to exploring small slices. So in 1980 he leapt ahead
to the thousand billionth Riemann zero and began calculating in that vicinity.
Still, the cliff face was perfectly flat.
At this point Odlyzko was calculating with 12-digit numbers, plus all the decimal places. Just to do the calculations, he had to invent new computational algorithms. At the time, he says, 1012 was “the highest order of zeros that we could reach with the calculation methods available to us.” Still, he suspected it wasn’t high enough. “The wildness of the zeta function grows extremely slowly,” Odlyzko tells me. Only when the “wildness” comes to the fore are deviant zeros likely to reveal themselves. In a sense, Odlyzko is searching for monsters, aberrations of the normal taxonomy whose location — like all good mythical beasts — is off the edge of any current map.
A decade later, he and a colleague invented a much better algorithm, and in one mighty bound Odlyzko leapt up the critical line to the 1020 Riemann zero. Now he was operating in a zone where no human had ever set foot. “To calculate all the zeros up to this point would require more computational time than there’s been in the history of humanity,” Odlyzko notes. Another jump brought him to the 1021 zero, then the 1022, and finally the 1023. At each step, billions more zeros fell to his computational scythe. But still the wild things eluded him.
“If I could do 1050, I would do it,” Odlyzko says. Yet he now suspects that even this would not be good enough. So slowly does the “wildness” grow, he believes that if exceptions exist off the critical line, they will probably not be found until around the 10100 Riemann zero. At present that region is beyond any currently conceivable computing power. And even if he could reach that high, Odlyzko notes philosophically that “counterexamples are likely to be extremely rare.” His chances of finding one are essentially nil. Nonetheless, he keeps on climbing.
In the mathematical universe, Odlyzko is applying what is known as a brute-force approach — the more computer power he can bring to bear, the more zeta zeros he can calculate, hence the more likely he is to find aberrations. On the whole, mathematicians disapprove of computational approaches; a widespread attitude, especially among older mathematicians, holds that the only “real” proof is an analysis derived from fundamental axioms. All else is hack work.
Odlyzko is aware of this bias among his colleagues, but he likens himself to an explorer venturing into a new land. “I am really going out there and looking at this wild universe and finding things that I hope will eventually lead to proofs,” he says. He is just now beginning to analyze his mammoth cache of zeros. “We simply don’t know what surprises the data might hold,” he declares. Odlyzko notes that this explorational view of mathematics is very much part of the subject’s tradition. Indeed, a marvelous new book by Amir Alexander, Geometric Landscapes (Stanford University Press), traces the history of the idea of the mathematician-explorer and shows how the rhetoric of discovery was integral to the way in which mathematicians of the scientific revolution conceived of themselves and their work.
Already Odlyzko’s forays into the stratospheric zone of the Riemann zeros have verified something astonishing. It turns out these zero points are not arranged randomly on the critical line. Mysteriously, they follow the same statistical pattern that physicists have found in some kinds of atomic systems — specifically, what are known as “quantum chaotic systems.” Thus, what seems at first a purely abstract discovery has turned up in nature. Nobody has the slightest idea why this might be so. But the revelation suggests the incredible possibility that we might be able to find (or build) a quantum system — perhaps some bizarre kind of atom — that would prove the Riemann Hypothesis. A number of physicists are now working toward that goal.
The interplay between mathematics and the material world has fascinated philosophers and scientists alike. “God ever geometrizes,” Plato declared. “All is number,” Tierry of Chartres concurred in the Middle Ages. Riemann himself developed his radical non-Euclidean geometry because he was convinced there must be a geometric explanation for the force of gravity. Fifty years after his death, Einstein demonstrated the truth of that insight. The link between Riemann’s zeta zeros and quantum mechanics suggests that understanding these zeros will help to illuminate the deeper mysteries of atoms, molecules and atomic nuclei.
Though Riemann’s Hypothesis was originally stated merely as an aside, it has turned out to be one of the most profound mathematical statements ever uttered. The deeper mathematicians go into it, the more connections they continue to discover. As Sabbagh writes, “The Riemann Zeta Function extends its tentacles into so many branches of mathematics it’s impossible to say where a solution might come from.” After so many years on the cliff face, no one has a greater investment in the problem than Odlyzko. I ask him if he thinks it will be resolved in his lifetime. Before answering, he pauses and on the other end of the phone I can hear a slow intake of breath. Yet his answer, when it comes, is full of optimism: “For all we know, it may have been done yesterday,” Odlyzko says. “It may be done tomorrow.”
Then again, he adds, “It may take another hundred years.”