Hilbert–Pólya conjecture

Establish the existence of a self-adjoint operator H such that the eigenvalues of 1/2 + iH coincide with the imaginary parts of the nontrivial zeros of the Riemann zeta function, thereby implying the Riemann hypothesis.

Background

The work situates its numerical experiments within the broader context of spectral approaches to the Riemann Hypothesis, notably the Hilbert–Pólya conjecture, which posits a deep operator-theoretic underpinning of the nontrivial zeros.

Demonstrating or constructing such a self-adjoint operator would provide a pathway to proving the Riemann Hypothesis by linking the zeros to a quantum-mechanical spectrum.

References

According to the Hilbert - Pólya conjecture; if H is a self-adjoint operator and the eigenvalues of 1/2 + iH correspond to the nontrivial zeros of the Riemann zeta function, then the Riemann hypothesis follows.

— Successive generation of nontrivial Riemann zeros from a Wu-Sprung type potential  (2510.16759 - Jaksch, 19 Oct 2025) in Introduction

The Hilbert-Pólya conjecture (1910s) suggests the existence of a self-adjoint operator $H$ such that the non-trivial solutions of $\zeta(1/2 + it) = 0$ are the eigenvalues of $H$.

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