The Hilbert-Pólya conjecture proposes that the nontrivial zeros of the Riemann zeta function can
be written 
,
where the values 
are the eigenvalues of a suitable
self-adjoint operator (Derbyshire 2004, pp. 277-278).
Since the eigenvalues of a self-adjoint operator are real, such a spectral correspondence would force 
and therefore prove the Riemann
hypothesis. No operator having all the required properties is known.
The conjecture motivates comparisons between the statistics of zeta zeros and spectra in quantum mechanics. Berry and Keating (1999ab) proposed the classical Hamiltonian
as a semiclassical model, while Connes (1999) formulated a trace-formula interpretation
in which the zeros occur as an absorption spectrum. These constructions give important
spectral analogies, but neither constitutes a proof of the conjecture or of the Riemann
hypothesis.
The Hilbert-Pólya conjecture is unrelated to the refuted number-theoretic Pólya conjecture concerning the Liouville function.
and the Riemann Zeros." In Supersymmetry and Trace
Formulae: Chaos and Disorder (Eds. I. V. Lerner, J. P. Keating,
and D. E. Khmelnitskii). New York: Plenum, pp. 355-367, 1999a.Berry,
M. V. and Keating, J. P. "The Riemann Zeros and Eigenvalue Asymptotics."
SIAM Rev. 41, 236-266, 1999b.