October 5, 2008

Quantum Physics May Offer Clues to Solving Prime Number Problem

By Castelvecchi, Davide

Numbers Electron energy levels linked to Riemann hypothesis

If two physicists are right, a single electron might know more about numbers than all of the world's mathematicians. In an upcoming Physical Review Letters, the researchers hint that the dynamics of an electron can embody the solution to the nearly 150-year-old Riemann hypothesis, a crucial unsolved problem that has wide and deep consequences for number theory.

German Sierra of the Spanish National Research Council in Madrid and Paul Townsend of the University of Cambridge in England propose that when an electron is confined to moving in two dimensions, its possible energy level values might encode the key to the hypothesis.

"They have gone a step forward toward giving a physical description of the Riemann hypothesis," says Jonathan Keating of the University of Bristol in England. He warns, though, that the problem may not have gotten easier as a result.

The hypothesis, or conjecture, was proposed by German mathematician Bernhard Riemann in 1859. It is regarded as important in large part because proving it would help reign in the apparent chaos in the world of prime numbers - whole numbers, such as 2, 3, 5, 7, 11 and so on, that can't be wholly divided by any numbers except 1 and themselves.

The hypothesis also has a $1 million "wanted" sign: The Clay Mathematics Institute in Cambridge, Mass., has offered a cash prize in exchange for the proof.

Mathematicians, at least since Euclid, have known that the list of prime numbers is infinite. But only one pattern has ever emerged from this list of primes. The prime number theorem, proved in the late 1800s, describes how primes become less frequent among larger numbers.

Roughly, it says that from one to 1 million (or 10^sup 6^), about one in every six numbers is prime; between one and 1 billion (or 10^sup 9^), it's about one in every nine. In general, between one and 10n, about one number for every n is a prime. (The actual statement includes a correction factor but is similar in spirit.)

At first sight, the Riemann hypothesis has nothing to do with prime numbers. It is a conjecture about a formula called Riemann's zeta function, which calculates a number for every point on a plane. Riemann's intuition was that the "zeros" of the function - points where zeta calculates the value zero - can lie along one of only two straight lines on the plane.

Mathematicians have shown that if the hypothesis is true, it would bolster the prime number theorem, implying there are no wild statistical fluctuations in the distribution of primes. While primes would still be unpredictable, complete chaos wouldn't rule.

Researchers have long suspected that there might be a way to convert the Riemann hypothesis into an equation similar to those used in quantum physics. The zeros of the zeta function could then be calculated the same way physicists, for example, calculate the possible energy levels for an electron in an atom.

Building on the ideas of Keating and others, Sierra and Townsend make that connection a bit more concrete. They suggest that an electron constrained to move in two dimensions, and subject to electric and magnetic fields, might have energy levels that match the zeros of the zeta function.

Demonstrating the existence of such a system, even on paper, would confirm the Riemann hypothesis. The physicists haven't quite done that, though. Their explicit model gives only an approximation of the energy levels they needed.

In the opinion of mathematician Enrico Bombieri of the Institute for Advanced Study in Princeton, N.J., the paper constitutes modest progress. He says physicists still haven't demonstrated a true connection between the function and physics. Until then, he adds, "attempts of this type belong to the works based on 'wishful thinking,' or even 'pie in the sky.'"

Keating, however, is more optimistic. "Maybe it will suggest further developments in the subject," he says.

The Riemann hypothesis says the zeta function cannot be zero except for along two particular lines on the plane. The points where all colors converge Into color wheels show some known zeros.

Copyright Science Service, Incorporated Sep 27, 2008

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