My friend Dan Rockmore has a new book out, entitled Stalking the Riemann Hypothesis, which is quite good. Dan had the misfortune of starting work on this book at the same time as several other people had the idea of a popular book about the Riemann Hypothesis. For better or worse, his has appeared after the others, which came out last year. In solidarity with him, I haven’t read the others, so can’t directly compare his to theirs.
Dan’s book begins with a mixture of history and explanations of the math involved. In the sections having to do with more recent work, he concentrates on one particular approach to proving the Riemann hypothesis, an approach that has interesting relations to physics. This involves an idea that goes back to Hilbert and Polya, that one should look for a quantum mechanical system whose Hamiltonian has eigenvalues given by the Riemann zeta-function zeros. Self-adjointness of the Hamiltonian then corresponds to the Riemann Hypothesis. This conjecture has motivated a lot of the research that Dan describes in detail, including relations to random matrix theory, quantization of chaotic dynamical systems, and much else.
Philosophically, I’m very fond of the idea that quantum mechanics is basically representation theory, and that the way to produce interesting quantum mechanical systems is by using geometric constructions of representations using cohomological or K-theoretic methods. While I’m no expert on the Riemann Hypothesis, my favorite idea about it is that proving it will require a mixture of the Hilbert-Polya search for a quantum mechanical system, together with the cohomological approach that worked in the case of function fields. In that case, the Weil conjectures famously were based on the idea of constructing an appropriate cohomology theory. This was carried through by Grothendieck and others during the fifties and sixties, with Deligne finally using this technique to get a proof in the early seventies.
For the number field case, the most developed conjecture that I know of about what might be the right sort of cohomology theory is due to Christopher Deninger. He has a very interesting recent review article about this, see also his lecture at the 1998 ICM.
Update: For another nice discussion of zeta-functions and the Riemann Hypothesis, see John Baez’s latest This Week’s Finds.
Update: There’s a nice article in the Washington Post about Dan and his book.