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Wednesday, July 03, 2002


After all the stir caused by Wolframs rule 110, here's another puzzle about our universe whose understanding might help us get things a little clearer: Riemann's Zeta function.

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......
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..............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.

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.

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.

Source: New York Times

The problem is to know whether this zeta function for primes has any significance, and if so, what is it?

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