## Physics of the Riemann Hypothesis

18/01/2011

In this blog I discuss frequently about one of the Clay Institute’s Millenium Prize problems: Mass gap and existence of a quantum Yang-Mills theory. Sometime I also used the Perelman’s theorem containing Poincarè’s conjecture to discuss about some properties of quantum gravity and also Cramer-Rao statistical bound. Today on arxiv I have found a beautiful review paper by Daniel Schumayer and David Hutchinson about Riemann hypothesis, another Millenium problem, and physics (see here). This question remained unsolved for almost 150 years since now. The relevance of the understanding of this conjecture relies on the possibility to give a function decribing the distribution of prime numbers.

The formulation of Riemann hypothesis is embarassingly simple. Riemann function is defined in a very simple way as

$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}.$

This function has a set of trivial zeros at all even negative integers and a set of nontrivial zeros. Riemann hypothesis claims that

All nontrivial zeros of $\zeta(s)$ have the form $\rho=\frac{1}{2}+it$, being t a real number.

This is the eighth problem of Hilbert that gave also the name we are using today to this question. Simple as may seem the question, it baffled mathematicians efforts since today. But, as happens to most mathematics, it can be found applied in Nature and it is tempting to think to reproduce in a lab what appears a complicated mathematical problem and read the answer directly from experiments. Indeed, such a road was definitely open in 1999 when Michael Berry (the one of the phase) and Jon Keating put forward an important conjecture relating quantum systems and Riemann hypothesis. You can find this cornerstone paper here. But since then the hunt was open to find other connections amenable to a treatment in physics. Schumayer and Hutchinson give an extensive review of them in their paper. This view opens up the possibility of a solution through physics of this fundamental question. Surely, we are assisting again at an interesting interwining between these fundamental disciplines of science.

Daniel Schumayer, & David A. W. Hutchinson (2011). Physics of the Riemann Hypothesis arxiv arXiv: 1101.3116v1

Berry, M., & Keating, J. (1999). The Riemann Zeros and Eigenvalue Asymptotics SIAM Review, 41 (2) DOI: 10.1137/S0036144598347497