Hilbert–Pólya conjecture

Construct a self-adjoint operator H whose spectrum equals the ordinates of the nontrivial zeros of the Riemann zeta function, thereby implying the Riemann Hypothesis.

Background

The conjecture proposes a spectral interpretation of the zeros of ζ(s) as eigenvalues of a self-adjoint operator, which would force them to be real (on the critical line after normalization).

This spectral perspective has inspired numerous approaches linking analytic number theory and operator theory, but no such operator is known.