Physics of the Riemann Hypothesis

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 we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics 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 hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.Comment: 27 pages, 9 figure

Mixmaster: Fact and Belief

We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state. Surprisingly few facts are known about the `Mixmaster' dynamics of these models, while at the same time most of the commonly held beliefs are rather vague. In this paper, we use Mixmaster facts as a base to build an infrastructure that makes it possible to sharpen the main Mixmaster beliefs. We formulate explicit conjectures concerning (i) the past asymptotic states of type IX solutions and (ii) the relevance of the Mixmaster/Kasner map for generic past asymptotic dynamics. The evidence for the conjectures is based on a study of the stochastic properties of this map in conjunction with dynamical systems techniques. We use a dynamical systems formulation, since this approach has so far been the only successful path to obtain theorems, but we also make comparisons with the `metric' and Hamiltonian `billiard' approaches.Comment: 34 pages, 10 figure

Planck pre-launch status : The Planck mission

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