By Christian Viveros-Fauné
By Miriam Felton-Dansky
By Tom Sellar
By Tom Sellar
By Jessica Dawson
By Tom Sellar
By R. C. Baker
By Tom Sellar
It remains unconquered to this day. Nash's crash-and-burn was more dramatic than mostdistracted and mumbling, he incoherently linked the hypothesis to arcane cosmology. But he is just one in a long line of formidable thinkers to have unsuccessfully tackled this conjecture, formulated by Bernhard Riemann in 1859 in a brief paper titled "On the Number of Prime Numbers Less Than a Given Quantity." In 1900, mathematician David Hilbert, addressing a conference in Paris, outlined what he judged the greatest unsolved problems in mathematics. Of the so-called Hilbert Problems (discounting the handful later deemed too vague for decisive solutions), the only one still standing at the start of the 21st century is the Riemann Hypothesis. In 2000, the Clay Mathematics Institute in Cambridge, Massachusetts, threw down the gauntlet again, announcing seven Millennium Problemsamong them the R.H.each one with a million-dollar prize attached.
Math conundrums have been known to languish unsolved for a seeming eternity. Long considered the holy grail, Fermat's Last Theorem (another Hilbert Problem) had stumped all comers for some 350 years before Andrew Wiles finally cracked it in 1994. There has been a flurry of RH research activity recently, spurred more by Wiles's triumph than the cash prize (mathematicians agree that immortal glory awaits the person who provesor refutesthe hypothesis). "People would never publicly acknowledge that they were working on the Riemann Hypothesis," says Dorian Goldfeld, a professor of mathematics at Columbia. "Now everyone's coming out of the closet." Brian Conrey, director of the American Institute of Mathematics in Palo Alto, California, writes in the March issue of Notices, the journal of the American Mathematical Society, "Now is arguably the most exciting time in its history to be working on R.H."
This month sees the publication of no fewer than three popular-math books focusing on primes and the hypothesis: Marcus du Sautoy's The Music of the Primes (HarperCollins), Karl Sabbagh's The Riemann Hypothesis (Farrar, Straus and Giroux), and John Derbyshire's Prime Obsession (Joseph Henry Press). While Wiles's proof made the front page of the Times, inspiring a bestseller (Simon Singh's Fermat's Enigma) and even an Off-Broadway musical (Fermat's Last Tangosample lyric: "Your proof contains a big fat hole"), it's questionable if the Riemann Hypothesis has similar crossover potential, not least because it still lacks a happy ending. At first glance, Fermat's is straightforward enough to seduce mathophobes; it simply states that there are no whole-number solutions of the equation xn+ yn = zn for n > 2. (The form will be familiar to anyone with even the faintest recollection of the Pythagorean Theorem.) By comparison, even a perfunctory definition of the Riemann Hypothesis necessarily entails an infinite series called the zeta function (1 + 1/2s + 1/3s + 1/4s + . . . ), which, for certain values of s, has a finite answer, and the daunting abstraction of complex numbers, which consist of real and imaginary parts.
From a mathematical standpoint, though, a Riemann proof would be more significant than the Fermat solution. It would shed light on the most basic objects in mathematics: prime numbers. A prime is a number perfectly divisible only by itself and the number 1; while every other number can be expressed as the product of at least two primes, the primes themselves are irreducible (11, 89, and 6,971, for instance, are prime; 15, which can be written as 3 x 5, is not). Pick your science metaphor: The primes are the atoms, the periodic elements, the genetic code of mathematics, and it frustrates mathematicians to no end that something so fundamental should remain so intractably mysterious. Primes seem to be randomly strewn amid the positive integers. Since their sequence conforms to no discernible formula, it's impossible to tell when the next prime will pop up.
Despite their chaotic behavior, as du Sautoy writes, the primes have a "timeless, universal character. Prime numbers would be there whether we had evolved sufficiently to identify them." In Carl Sagan's novel Contact (1985), scientists realize that the messages being transmitted from Vega are prime-number pulsesand thus signs of intelligent life. In discussing the primes, mathematicians often use the vocabulary of first love. "They're objects of great beauty, no question," says AIM's Conrey, explaining how as a teenager he fell for the Twin Prime Conjecture, which states that there are an infinite number of pairs of primes separated by 2, like 3 and 5, 17 and 19 (it's still unproved).
For a vivid picture of prime fetishism, check out Chris Caldwell's encyclopedic and surprisingly addictive Prime Pages (www.utm.edu/research/primes). Among the contents: a list of the largest known primes (updated weekly), a primality test for numbers smaller than 253 - 1, and abundant prime trivia ("17 was the original title of the Beatles song 'I Saw Her Standing There' "). Over at the Aesthetics of the Prime Sequence (2357.a-tu.net), a counter reels off primes as you listen to spooky minimalist compositions derived by feeding prime numbers into an algorithm, or take a prime intuition test (a crude de facto autism test, perhapsOliver Sacks once recounted the bizarre case of autistic twin brothers whose private language consisted of exchanging primes of up to 20 digits).
Prime number theory dates to 350 B.C., when Euclid proved that there are an infinite number of primes. But it was Karl Friedrich Gauss, a future teacher of Riemann's, who in the late 18th century laid the groundwork for the latter's breakthrough. At 14, Gauss observed that while individual primes are randomly dispersed, there is an overall pattern to their density: Not only do they thin out the higher you count, the proportion of primes seems to follow a logarithmic function. (For Gauss, according to Sabbagh's book, counting primes was a pastimeit's said that by the time he died, he'd gotten to 3 million.) His conjecture, that the number of primes less than any given number n is approximately n/log n (proved more than a century later in 1896), is called the Prime Number Theorem.
Gauss's formulation provided only an estimate. Riemann, using the zeta function, derived an exact formula for the number of primes smaller than any given number. The Swiss mathematician Leonhard Euler had discovered that the zeta function, which we defined above as an infinite sum involving all the natural numbers (1, 2, 3, . . . ), could also be expressed as an infinite product involving just the prime numbers. This was, as du Sautoy writes, "the first sign that the zeta function might reveal unexpected links between seemingly disparate parts of the mathematical canon." Riemann's insight was to plug complex numbers into the zeta functioncomplex numbers have real and imaginary parts, the latter being numbers that can be expressed in terms of i, where i is the square root of -1.
These suspicious-sounding numbers may seem like a flight of mathematical fancythey were first "imagined" in the 16th century and found to have enormous computational significance, but even mathematicians needed a few hundred years to wrap their heads around the concept, just as it took the ancient Greeks a while to get used to the initially horrifying idea of fractions. (Barry Mazur's recent Imagining Numbers offers an eccentric road map through this abstract terrain.)
Riemann relocated the enigma of prime distribution to the complex plane (a 2-D plane with the x-axis real and the y-axis imaginary). This magical shift, which du Sautoy likens to a beyond-the-looking-glass maneuver, revealed an astonishing regularity. The complex numbers for which the zeta function equals zero are called the Riemann zeros; the frequency of the Riemann zeros mirrors the frequency of the primes. Riemann's famous hypothesis, which he declared sehr wahrscheinlich ("very probable"), was that, mapped on the complex plane, these solutions fall on a straight lineto be exact, the vertical line that runs through the point 1/2 on the real axis.
Convoluted as it sounds, this gloss on the problem grossly simplifies the heady mathematics at its core. All three new books are careful not to outpace the general reader. Du Sautoy, a professor at Oxford, provides a panoramic history of prime-number crunching, rich with anecdote and unfailingly patient with the mathematical fine points. Sabbagh, a documentary producer, uses the R.H. to tunnel into the mathematical mind, folding in copious interviews with number theorists. His approach to the math is at once reassuring and deflationary. Chapter 16 opens: "After fifteen chapters of a book on the Riemann Hypothesis, I have to break some bad news to you. You know almost nothing about the Riemann Hypothesis compared with what there is to know." Derbyshire, a National Review columnist, has written the most mathematically detailed of the trio. Those looking for a quick, lucid R.H. initiation should consult Keith Devlin's evocative chapter in The Millennium Problems (Basic Books, 2002), which also takes on the other six Clay puzzles.
Like many other conjectures, the Riemann Hypothesis could conceivably be disproved by brute tenacityrepeatedly testing it until a counterexample is found. Indeed, the R.H. has been checkedand holds truefor the first 100 billion Riemann zeros. At the AT&T Labs in Princeton, supercomputers keep cranking out zeros, an alarm ready to go off if and when the first zero strays from the so-called critical line.
You'd think this would count as overwhelming evidence in favor of the hypothesis, but mathematics is a science of absolutes, insisting on the concreteness of universal truth as if to compensate for some of its more vaporous abstractions. Du Sautoy says in an e-mail interview, "For the other sciences, evidence in the lab is everything. In the mathematician's lab, such evidence can be completely misleading." In other words, extrapolation is a fool's game when you're dealing with infinity. "The primes are a malicious bunch of numbers," du Sautoy continues. "They're the masters of disguise. People have made conjectures about primes supported by billions of pieces of computer evidence, but theoretical analysis has later revealed the evidence to be a charade."
There is the notorious failure of Merten's Conjecture, proved false in 1983 after being verified for the first 10 billion cases. Columbia's Goldfeld points to the early-20th-century insights into Gauss's P.N.T. estimate, which was long assumed to be an overestimate. The English mathematician J.E. Littlewood proved in 1914 that at some point Gauss's formula becomes an underestimate. Two decades later Stanley Skewes proved that this happens before
Ten to the 10th power, to the 10th power, to the 34th powera strong contender for the most meaningless result in mathematics, since Skewes's number, as it would be called, is so preposterously huge as to be beyond human comprehension (there are fewer atoms in the universe, Goldfeld points out). This "upper bound" has since been reduced to 10371, but the moral is that the AT&T computers could go on firing forever, and the eternal absence of any delinquent zeros would not in itself make for a proof.
Most obviously, a proof of the R.H. would have a domino effect on numerous subsequent theorems that begin by assuming its truth. (The turmoil that a false verdict would wreak on their field is something mathematicians prefer not to think about.) But a proof will mean more than simple verification. "It's not just about knowing, yeah, the R.H. is true," explains Conrey. "If that was all, we might as well assume it right now and go about our business, because it probably is true." The hope is that something fundamental will be uncovered, since, as Conrey says, "the Riemann zeta function represents a connection between addition and multiplication that we don't yet understand." A proof could be the first step toward a grand unified field theory. "The history of mathematics can be seen as the evolution of a language to articulate structure and patterns," du Sautoy says. "Many feel that we still haven't developed the right language to articulate why the Riemann Hypothesis is true." As Goldfeld puts it, "With a problem that old, solving it usually means having to invent a whole new type of mathematics. You're not going to do it with a little hat trick."
Parallel mysteries are being toppled all the time. In the most important recent breakthrough, Indian computer scientist Manindra Agrawal and two of his students stunned mathematicians last August when they unveiled a relatively straightforward algorithm that would test if any given number is prime. Just two weeks ago, AIM announced a major number theory development by Dan Goldston and Cem Yildiriman approach toward "small gaps between consecutive primes" that could help prove Conrey's beloved Twin Prime Conjecture.
In real-world terms, an R.H. proof would likely have massive implications for cryptography. The security of Internet transactions relies on the virtual impossibility of factoring extremely large numbers; a public-key encryption system deploys two large primes (kept secret, used for decryption) and their product (public knowledge, used for encryption). Agrawal's primality test sent a wave of anxiety rippling through the world of cryptography, suggesting that a discovery in the related area of factoring could be imminent. Du Sautoy compares the elusive factoring algorithm to a spectrometer, which breaks down molecules into their composite atoms. "Since the Riemann Hypothesis holds the prospect of revealing the secret harmony underlying the primes," he adds, "its proof might provide insight into how to build a prime-number spectrometer. The Riemann Hypothesis could bring e-business crashing to the ground."
Some believe a proof could have literally cosmic significance. The chance meeting between mathematician Hugh Montgomery and physicist Freeman Dyson, over afternoon tea at Princeton in 1972, is a key moment in Riemannic lore. Dyson noticed that Montgomery's formula for the spacings between Riemann zeros corresponded to a formula for the spacings between energy levels for a chaotic system. Could quantum physics and number theory somehow be linked? (Maybe Nash's lecture on space-time singularities wasn't so far off.) In the '90s, French mathematician Alain Connes attempted to bridge the two fields by enlisting a new type of geometry involving infinite-dimensional spaces, though the extreme obscurity of his approach can be judged by just how many in the math community fell prey to the April Fool's e-mail that circulated in 1997 claiming that a young physicist, after attending a Connes lecture, had quickly dashed off a proof based on, among other things, "anyons and morons with opposite spins."
AIM, co-founded by Silicon Valley mogul John Fry as a nonprofit organization for collaborative mathematics, leads the way in Riemann research. Even though Wiles famously worked on Fermat's in isolation (and the cash prize is surely an incentive to hole up by yourself), Conrey says the response to the institute's R.H. symposia has been enthusiastic: "We've sped things up a bit. There are a lot more people thinking about it now."
After 140-plus years, is a solution in sight? Is it even possible to tell? Goldfeld expresses cautious optimism: "I think it'll be cracked in 10 to 20 years." But, he adds, "most of the approaches to the Riemann Hypothesis do seem too close to the problem. What we need is someone who'll connect it with something really different." Conrey says he's most encouraged by the work of Rutgers professor Henryk Iwaniec in the area of Möbius functions. It's worth noting that according to Kurt Gödel's Undecidability Theorem (an elaboration of the liar's paradox), some statements are true but can never be proved. (Indeed, one of the Hilbert problems was proved to be unprovable.) Riemann never claimed his conjecture was provable, unlike Fermat, who left a taunting note in the margin, saying he'd run out of space for his "truly marvelous" proof. Still, there are hardly any mathematicians working today who believe the Riemann Hypothesis to be false or unprovable, and for many of them, keeping the faith is practically a religious imperative. "Aesthetics is the root of most people's belief that the Riemann Hypothesis will turn out to be true," says du Sautoy. "We're blessed with a subject where nature invariably chooses the most beautiful option."