Hamiltonian for the zeros of the Riemann zeta function

TL;DR

This paper constructs a Hamiltonian operator whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function, offering a potential physical framework to address the Riemann hypothesis.

Contribution

It introduces a non-Hermitian Hamiltonian with PT symmetry related to the Riemann zeros and discusses conditions under which it could be self-adjoint, linking quantum mechanics to number theory.

Findings

Hamiltonian linked to Riemann zeros

PT symmetry allows real eigenvalues

Potential path to proving the Riemann hypothesis

Abstract

A Hamiltonian operator $\hat H$ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of $\hat H$ is $2xp$, which is consistent with the Berry-Keating conjecture. While $\hat H$ is not Hermitian in the conventional sense, ${\rm i}{\hat H}$ is ${\cal PT}$ symmetric with a broken ${\cal PT}$ symmetry, thus allowing for the possibility that all eigenvalues of $\hat H$ are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that ${\hat H}$ is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.