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143-Year-Old Problem Still Has Mathematicians Guessing
By BRUCE SCHECHTER

n the early years of the 20th century, the great British mathematician Godfrey Harold Hardy used to take out a peculiar form of travel insurance before boarding a boat to cross the North Sea. If the weather looked threatening he would send a postcard on which he announced the solution of the Riemann hypothesis. Hardy, wrote his biographer, Constance Reid, was convinced "that God - with whom he waged a very personal war - would not let Hardy die with such glory."

The Riemann hypothesis, first tossed off by Bernhard Riemann in 1859 in a paper about the distribution of prime numbers, is still widely considered to be one of the greatest unsolved problems in mathematics, sure to wreath its conqueror with glory - and, incidentally, lots of cash. Two years ago, to celebrate the millennium, the Clay Mathematics Institute announced an award of a million dollars for a proof (or refutation) of the hypothesis.

Whether in pursuit of glory, cash ("prizes attract cranks," one mathematician sniffed) or pure mental satisfaction, more than a hundred of the world's leading mathematicians came to New York City recently to attend an unusual conference at New York University's Courant Institute. While most math conferences are devoted to presenting completed work, this one was held for mathematicians to swap hunches, warn of dead ends and get new ideas that could ultimately lead to a solution.

"One of the things we hope to do is to consolidate the approaches," said Dr. Brian Conrey, a professor of mathematics at Oklahoma State University and executive director of the American Institute of Mathematics, a private group that organized the meeting with support from the Courant Institute and the National Science Foundation. "We're looking for brand-new ideas with which to open the door."

There was a guarded optimism among the mathematicians that promising new ideas were being put forward, but in mathematics prognostication is a dangerous game. Hardy, for example, rated the Riemann hypothesis less difficult than Fermat's conjecture, which Dr. Andrew Wiles of Princeton solved in 1993, after working for seven years in secrecy. Dr. Wiles, as it happens, dropped in on the conference, but when asked if this meant he was now attacking the hypothesis he shrugged and said, "Well, it's a spectator sport, you know."

As in all sports, it helps to know the rules of the game. Riemann made his hypothesis in the course of a 10-page paper he wrote on the distribution of prime numbers that is considered to be one of the most important papers in the history of number theory, a history that stretches back more than 2,500 years.

Prime numbers are numbers that are divisible only by one and themselves - they are the atoms of arithmetic, for any number is either a prime or a product of primes. The first few primes are 2, 3, 5, 7, 11 and 13 - but despite their simple definition the prime numbers appear to be scattered randomly amid the integers.

There is no simple way to tell if a number is prime, and that is the basis for most modern encryption schemes. Solving the hypothesis could lead to new encryption schemes and possibly provide tools that would make existing schemes, which depend on the properties of prime numbers, more vulnerable.

Despite the random occurrence of individual primes, the primes themselves were found to follow a remarkably simple distribution. In 1792, when he was 15, Karl Friedrich Gauss decided to examine the number of primes less than a given number. He discovered that the primes became, on average, sparser the further out he looked and that this dwindling obeyed a simple, logarithmic law. He had no idea why this was so, but it was intriguing.

In 1859, Riemann, who had been a student of Gauss, took up the question of the distribution of primes in his only paper on number theory. With that paper he revolutionized the field, as he had the fields of geometry (his math became the basis for Einstein's theory of gravitation) and several other branches of mathematics. What Riemann discovered was a way of using the properties of a relatively simple function to count the primes.

What was so remarkable about Riemann's zeta function was that it somehow took a question about prime numbers - those discrete atoms of simple arithmetic, things easy to imagine - and put it in terms of a far larger and more esoteric class of numbers known as complex numbers. Complex numbers are a generalization of the familiar decimal numbers that mathematicians call the real numbers.

While the real numbers can be thought of as points on an infinite line, the complex numbers are points on a plane. One axis of this complex plane corresponds to the real numbers, and the other corresponds to the "imaginary" numbers - which were introduced so that negative numbers could have square roots, and are no more imaginary than real numbers. A function like Riemann's zeta function is simply a rule that takes a point on this plane and sends it to some other point.

By moving the problem to the complex plane Riemann had access to a whole new set of powerful mathematical tools, many of which he had developed himself. What was going on with the primes turned out to be a shadow of what was going on in this more general world.

Riemann showed that if he knew where the value of his zeta function went to zero he would be able to predict the distribution of the primes. He was able to prove that aside from some "trivial" zeros - located at -2, -4, -6, and so on and thus easily included in his equations - the zeros of the zeta function all lay within a strip one unit wide running along the imaginary axis.

Somehow the distribution of these zeros mirrored or encoded the distribution of the prime numbers. Riemann guessed that all of the zeros ran along the middle of the critical strip like the dotted line on a highway. Nobody is sure why he made this guess, but it has proven to be inspired. Over the past few decades billions of zeros of the zeta function have been calculated by computer, and every one of them obeys Riemann's hypothesis.

Most of the conference attendees would be shocked if a stray zero were found and Riemann was proven wrong. They would agree with John Frye, the chief executive of Frye's Electronics and a math major who used his fortune to found the mathematics institute. "I think we would have a better chance of finding life on Mars than finding a counter-example," he said.

But the field is rife with examples of hypotheses that seem to be true but are subsequently proven to fail at numbers beyond the reach of any conceivable computer. Only a mathematical proof, based on logic, can handle questions of the infinite.

Still, calculating the zeros of zeta is not an idle pursuit. In 1972, Hugh Montgomery, a mathematician at the University of Michigan, investigated the statistical distribution of the zeros. He found that they were scattered randomly but seemed to repel each other slightly - they did not clump together. On a trip to the Institute for Advanced Study in Princeton he showed his result to the physicist Freeman Dyson.

By sheer luck, Dr. Dyson was one of the few people in the world who would have recognized that the Montgomery results looked just like recent calculations on the energy levels of large atoms. The coincidence was so striking that it forged a new and still mysterious bridge between quantum physics and number theory. The connection was one of many pursued at the conference, though Dr. Montgomery does not think this work will lead directly to a solution. "It only gives us clues," he said.

Other clues abounded at the conference, some tantalizing, such as possible linkages to the theories Dr. Wiles developed to solve the Fermat conjecture. But mathematical proofs are extremely delicate structures that can vanish at the merest touch.

Dr. Peter Sarnak, from the Institute for Advanced Study, spoke to the meeting about a promising approach that he and his colleagues have been pursuing. Just as Riemann attacked the problem of the primes by generalizing the zeta function to the complex plane, Dr. Sarnak and many others have been looking at families of functions of which Riemann's zeta function is just one relative. Each of these functions has its own Riemann hypothesis. "Of course," Dr. Sarnak acknowledged, "often the reason you generalize is that you're stuck."

But generalization also has its rewards. While the Riemann hypothesis does not have very many applications, the generalized version, if true, would solve hundreds of important mathematical problems.

When Dr. Wiles sat down in his attack to solve Fermat's conjecture, his path, though it would require genius to traverse, was clear: recent results had indicated the most promising direction to travel. Mathematicians at the conference agreed that there was no such clear evidence of a trail head for the Riemann hypothesis, a challenge they called both frustrating and exhilarating.

"The Riemann hypothesis is not the last word about things," Dr. Montgomery said. "It should be the first fundamental theorem. We're in a kind of logjam right now because we can't prove the fundamental theorem."