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.