The Berry-Keating Hamiltonian and the Local Riemann Hypothesis

TL;DR

This paper links the zeros of the Mellin transform of harmonic oscillator eigenfunctions to the eigenvalues of the Berry-Keating Hamiltonian, providing a spectral proof of the local Riemann hypothesis over the reals.

Contribution

It establishes a novel spectral interpretation of the local Riemann hypothesis using the Berry-Keating Hamiltonian for the real case.

Findings

Zeros' imaginary parts are eigenvalues of the Berry-Keating Hamiltonian.

Provides a spectral proof of the local Riemann hypothesis over the reals.

Discusses the p-adic case briefly.

Abstract

The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.