Hilbert-Pólya Conjecture

The Hilbert-Pólya conjecture is the proposal that the non-trivial zeros of the Riemann zeta function correspond to the eigenvalues of some unbounded self-adjoint operator — an operator that would play the same role for the primes that the Hamiltonian plays for a quantum system. If true, the Riemann hypothesis would follow immediately, since the eigenvalues of a self-adjoint operator are real.

The conjecture originated in informal conversations between George Pólya and Edmund Landau in the 1910s, and was later attributed to David Hilbert as well. Despite nearly a century of effort, no such operator has been constructed. The difficulty is not merely technical. The operator, if it exists, would have to encode the full arithmetic structure of the integers — including their prime factorization — in its spectral properties, a requirement that no known physical system satisfies.

The conjecture remains the most direct bridge between analytic number theory and quantum mechanics, and its resolution would either unify these fields or demonstrate that their apparent convergence is a coincidence deeper than any proof. The statistical evidence — the Montgomery-Odlyzko law, the agreement between zeta zero spacings and random matrix eigenvalue spacings — is so precise that it has led to the stronger Berry conjecture, which predicts the exact form of the pair correlation function of the zeta zeros.