Hilbert–Pólya conjecture

In mathematics, the Hilbert–Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory.

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 an unbounded self-adjoint operator. The earliest published statement of the conjecture seems to be in Montgomery (1973).

David Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture for reasons that are anecdotal.

At the time of Pólya's conversation with Landau, there was little basis for such speculation. However Selberg in the early 1950s proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian. This so-called Selberg trace formula bore a striking resemblance to the explicit formulae, which gave credibility to the Hilbert–Pólya conjecture.

Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property, now called Montgomery's pair correlation conjecture. The zeros tend not to cluster too closely together, but to repel. 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.

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