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Colloquium: Physics of the Riemann hypothesis

Dániel Schumayer* and David A. W. Hutchinson

Dániel Schumayer* and David A. W. Hutchinson

  • Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin, New Zealand 9016

  • *dschumayer@physics.otago.ac.nz
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Rev. Mod. Phys. 83, 307 – Published 29 April, 2011Erratum Rev. Mod. Phys. 83, 769 (2011)

DOI: https://doi.org/10.1103/RevModPhys.83.307

Abstract

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here a particular number-theoretical function is chosen, the Riemann zeta function, and its influence on the realm of physics is examined and also how physics may be suggestive for the resolution of one of mathematics’ most famous unconfirmed conjectures, the Riemann hypothesis. Does physics hold an essential key to the solution for this more than 100-year-old problem? In this work numerous models from different branches of physics are examined, from classical mechanics to statistical physics, where this function plays an integral role. This function is also shown to be related to quantum chaos and how its pole structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations light is shed on how the Riemann hypothesis can highlight physics. Naturally, the aim is not to be comprehensive, but rather focusing on the major models and aim to give an informed starting point for the interested reader.

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Corrections

4 May, 2011

Erratum

Publisher’s Note: Colloquium: Physics of the Riemann hypothesis [Rev. Mod. Phys. 83, 307 (2011)]

Dániel Schumayer and David A. W. Hutchinson
Rev. Mod. Phys. 83, 769 (2011)

Article Text

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Rev. Mod. Phys. 83, 307– Published 29 April, 2011
  • Received 10 July 2008
  • Publisher error corrected 4 May 2011

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