The Riemann Hypothesis

Spectral Interpretation

One idea for proving the Riemann hypothesis is to give a spectral interpretation of the zeros. That is, if the zeros can be interpreted as the eigenvalues of 1/2+iT, where T is a Hermitian operator on some Hilbert space, then since the zeros of a Hermitian operator are real, the Riemann hypothesis follows. This idea was originally put forth by Polya and Hilbert, and serious support for this idea was found in the resemblence between the "explicit formulae" of prime number theory, which go back to Riemann and Von Mangoldt, but which were formalized as a duality principle by Weil, on the one hand, and the Selberg trace formula on the other.

GUE

The best evidence for the spectral interpretation comes from the theory of the Gaussian Unitary Ensemble, which shows that the local behavior of the zeros mimics that of a random Hamiltonian. The link gives a more extended discussion of this topic.

Goldfeld

Goldfeld gave two spectral interpretations of the zeros of the zeta function; neither of these seems to prove the Riemann hypothesis. For example, in one interpretation, the zeros are eigenvalues of an operator, but it is unclear why the operator should be Hermitian.

Connes

In Comptes Rendus Acad. Sci. Paris, t.323, Série I, 1231-1236, December 1996, Alain Connes offered an attack on the Riemann hypothesis. This paper, titled "Formule de trace en géométrie non-commutative et hypothèse de Riemann" has the following

Abstract: We reduce the Riemann hypothesis for L-functions on a global field k to the validity (not rigorously justified) of a trace formula for the action of the idele class group on the noncommutative space quotient of the adeles of k by the multiplicative group of k.

During the 1996 Seattle conference on the Riemann hypothesis, Connes gave a fascinating talk on the work of Connes and Bost, "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. 1 (1995), 411-457.

Papers at this Site

Here is a paper, A Local Riemann hypothesis describing how local Tate integrals formed with eigenfunctions of the quantum mechanical harmonic oscillator, and its p-adic analogs, have their zeros on the line re(s)=1/2. This version of the paper (changed November 7, 1996) incorporates new material on the harmonic oscillator in n dimensions, Mellin transforms of the Laguerre functions, and a reciprocity law for their values at negative integers.

• Lecture Notes from a class. (Not very polished or original.)

Note: If you are using Netscape, you may want to use Shift-Left Mouse Button to download the file.

RH Links

• Dave Goss' home page includes a paper on RH in characteristic p.
• De Branges' manuscript, claiming a proof.
• A critique of de Brange's proof from the preprint server at the American Institute of Mathematics (which has several other number theory papers).
• A nonlinear equation and its application to nearest neighbor spacings for zeros of the zeta function and eigenvalues of random matrices by Forrester and Odlyzko.
• Steve Finch's page on the de Bruijn-Newman constant.
• The home page of Michael Rubinstein, including his dissertation on the local statistics of the zeros of L-functions;
• Deninger
• Noncommutative Geometry, Trace formulas and the zeros of the Riemann zeta function, a course by Alain Connes, eventually to contain lecture notes;
• Marek Wolf
• Hypergeometric representation of the Zeta Function of Riemann by Krzysztof Maslanka.
• The L-function page L-function page of Brian Conrey and David Farmer.