Hamiltonian for the zeros of the Riemann zeta function

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, $\mathrm{i}\hat{H}$ is $\mathcal{P}\mathcal{T}$ symmetric with a broken $\mathcal{P}\mathcal{T}$ 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.

5 pages, version to appear in Phys. Rev. Lett

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