Synopsis: A Riemann calculator?

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Landau Levels and Riemann Zeros

Germán Sierra and Paul K. Townsend

Published September 12, 2008

Perhaps the greatest unsolved problem in mathematics is the Riemann hypothesis, which states that the nontrivial zeros of the zeta function all have a real part equal to 1/2. Many results in mathematics, through their relation to the distribution of prime numbers, are based on whether the hypothesis is correct.

One possible route to a proof of the hypothesis is to find a quantum mechanical system whose quantized energy levels yield the nontrivial zeros of the zeta function. About a decade ago, it was conjectured that the correct quantum system is related to a particular classical system that exhibits chaotic dynamics. Writing in Physical Review Letters, Germán Sierra from CSIC-UAM in Spain and Paul Townsend from the University of Cambridge extend part of this classical model to a realistic quantum mechanical system: a charged particle moving in a plane in a uniform magnetic field and a saddle-shaped electric potential. Although it is not a proof of the Riemann hypothesis, Sierra and Townsend’s idea makes an interesting connection between a physical system—similar to the one in which the quantum Hall effect was measured—and efforts to solve a long-standing mathematical problem. – Sonja Grondalski

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