By Christian Viveros-Fauné
By Miriam Felton-Dansky
By Tom Sellar
By Tom Sellar
By Jessica Dawson
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By R. C. Baker
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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."