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In mathematics, the Hilbert-Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory.
Hilbert and Pólya speculated that values of t such that 1/2 + it is a zero of the Riemann zeta function might be the eigenvalues of a Hermitian operator, and that this would be a way of proving the Riemann hypothesis. At the time, 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 speculation of Hilbert and Pólya.
Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property. The zeros tend not to be 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, which is of importance in physics — the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics.
Dyson saw that the statistical distribution found by Montgomery was exactly the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix. Subsequent work has strongly borne out this discovery, and the distribution of the zeros of the Riemann zeta function is now believed to satisfy the same statistics as the eigenvalues of a random Hermitian matrix, the statistics of the so-called Gaussian Unitary Ensemble. Thus the conjecture of Pólya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.
In a development that has given substantive force to this approach to the Riemann hypothesis through functional analysis, Alain Connes has formulated a trace formula that is actually equivalent to a generalized Riemann hypothesis. This has therefore strengthened the analogy with the Selberg trace formula to the point where it gives precise statements.