Colloquium: Physics of the Riemann hypothesis
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 numbertheoretical 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 100yearold 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 BoseEinstein 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.
 Publication:

Reviews of Modern Physics
 Pub Date:
 April 2011
 DOI:
 10.1103/RevModPhys.83.307
 arXiv:
 arXiv:1101.3116
 Bibcode:
 2011RvMP...83..307S
 Keywords:

 02.10.De;
 02.30.Gp;
 02.70.Hm;
 Algebraic structures and number theory;
 Special functions;
 Spectral methods;
 Mathematical Physics;
 Condensed Matter  Quantum Gases;
 Condensed Matter  Statistical Mechanics;
 High Energy Physics  Theory
 EPrint:
 27 pages, 9 figures