A quantum mechanical model of the Riemann zeros

TL;DR

This paper develops a quantum mechanical model that generalizes Berry and Keating's approach to represent Riemann zeros as resonances, incorporating boundary effects and symmetries related to the zeta function.

Contribution

It introduces a quantized xp Hamiltonian with boundary-dependent interactions, exactly solves the model, and links boundary wave functions to Riemann zeros and L-functions.

Findings

Average Riemann zeros become resonances in the model

The model's symmetry relates to the zeta function's duality

Dirichlet L-functions are naturally realized within the framework

Abstract

In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wave functions, associated to the Berry-Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wave functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model.