Berry–Keating conjecture

Develop a rigorous quantization of a classical chaotic Hamiltonian, notably H = 1/2(xp + px) (or H = xp), with appropriate boundary conditions, whose self-adjoint quantization has eigenvalues equal to the ordinates of the nontrivial zeros of the Riemann zeta function.

Background

The Berry–Keating conjecture suggests a concrete Hamiltonian framework for realizing the Hilbert–Pólya vision via quantization of the classical xp dynamics.

Although compelling connections to quantum chaos and random matrix statistics are known, a rigorous construction yielding exactly the zeta zeros is still missing.

References

The Berry-Keating conjecture suggests connections to classical chaotic Hamiltonian systems whose quantization might yield the desired operator. Specific proposals include $H =\frac 12( xp+px)$ (where $x$ is position and $p$ is momentum) with appropriate boundary conditions, though rigorous constructions remain elusive.

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Subsubsection Scattering theory and spectral interpretation