In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator.
In a letter to Andrew Odlyzko, dated January 3, 1982, George Pólya said that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts t of the zeros of the Riemann zeta function corresponded to eigenvalues of a self-adjoint operator.
[1][2] David Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture due to a story told by Ernst Hellinger, a student of Hilbert, to André Weil.
Hellinger said that Hilbert announced in his seminar in the early 1900s that he expected the Riemann Hypothesis would be a consequence of Fredholm's work on integral equations with a symmetric kernel.
However Selberg in the early 1950s proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian.
[2] Visiting at the Institute for Advanced Study in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices.
These distributions are of importance in physics — the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics.
Subsequent work has strongly borne out the connection between the distribution of the zeros of the Riemann zeta function and the eigenvalues of a random Hermitian matrix drawn from the Gaussian unitary ensemble, and both are now believed to obey the same statistics.
Thus the Hilbert–Pólya conjecture now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.
[7] In 1998, Alain Connes formulated a trace formula that is actually equivalent to the Riemann hypothesis.
This strengthened the analogy with the Selberg trace formula to the point where it gives precise statements.
Using perturbation theory to first order, the energy of the nth eigenstate is related to the expectation value of the potential: where
Such integral equations may be solved by means of the resolvent kernel, so that the potential may be written as where
Berry and Keating have conjectured that since this operator is invariant under dilations perhaps the boundary condition f(nx) = f(x) for integer n may help to get the correct asymptotic results valid for large n A paper was published in March 2017, written by Carl M. Bender, Dorje C. Brody, and Markus P. Müller,[11] which builds on Berry's approach to the problem.
There the operator was introduced, which they claim satisfies a certain modified versions of the conditions of the Hilbert–Pólya conjecture.