There’s a fascinating new preprint out from Alain Connes, called An essay on the Riemann Hypothesis, written for a volume on “Open Problems in Mathematics”. Evidently the late John Nash is an editor, and responsible for commissioning this piece.
Connes is a mathematician of the first rank, and a very original one at that. He has now struggled with the Riemann hypothesis for many years, and his account of various approaches to the problem and the state of efforts to pursue them is a remarkable document of a sort that too rarely gets written.
Much of what he is concerned with is the question of how to find a proof along lines related to those used to prove the analog of the Riemann hypothesis in the case of function fields (this was successfully carried out by Deligne in the early 1970s). James Milne has a wonderful expository piece on the topic of this proof, going into details of the history and the mathematics. It provides a great supplement to the more speculative article by Connes.
For something much more concrete about the Riemann hypothesis, there’s a new book by Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis. Among a long list of attempts to relate this to physics, there’s an interesting relatively recent discussion of one idea from John Baez.
Thanks for the link to the book by Barry Mazur and William Stein. Browsing the available PDF of the book, it strongly seems to me that this is a book I just have to buy when it arrives in the stores. What I like about it is its, at least apparently, accessibility for non-mathematicians (I am myself a physicist) and people not experts in the field of the Riemann Hypothesis.
I can’t resist to quote:
Physics of the Riemann Hypothesis arXiv:1101.3116
A little less known (e.g. not mentioned in reference by a 1) connection to physics and in particular QFT is that the Riemann Hypothesis is equivalent to the existence of a 1d QFT with specified translation invariant 2-point function. See page 6 of “The explicit formula in simple terms” by Jean-Francois Burnol http://arxiv.org/abs/math/9810169
or the earlier paper http://arxiv.org/abs/math/9809119
Here it is QFT in the Euclidean formulation and the main requirement is not unitary or Osterwalder-Schrader positivity but rather Nelson-Symanzik positivity which makes the model an honest probability distribution on D'(R) or D'((0,infinity)) if one wants to use the log or not.