Landau levels and Riemann zeros

TL;DR

This paper connects the zeros of the Riemann zeta function to Landau levels in quantum mechanics, proposing a model where the lowest Landau level relates to the smooth counting function of zeros, and higher levels relate to fluctuations.

Contribution

It demonstrates how Connes' absorption spectrum model arises as the lowest Landau level limit of a quantum system, linking quantum physics to the distribution of Riemann zeros.

Findings

Lowest Landau level corresponds to the smooth part of N(E)

Higher Landau levels may account for fluctuations in N(E)

Quantum model provides a physical interpretation of Riemann zeros

Abstract

The number $N(E)$ of complex zeros of the Riemann zeta function with positive imaginary part less than $E$ is the sum of a `smooth' function $\bar N(E)$ and a `fluctuation'. Berry and Keating have shown that the asymptotic expansion of $\bar N(E)$ counts states of positive energy less than $E$ in a `regularized' semi-classical model with classical Hamiltonian $H=xp$. For a different regularization, Connes has shown that it counts states `missing' from a continuum. Here we show how the `absorption spectrum' model of Connes emerges as the lowest Landau level limit of a specific quantum mechanical model for a charged particle on a planar surface in an electric potential and uniform magnetic field. We suggest a role for the higher Landau levels in the fluctuation part of $N(E)$.