Landau Levels and Riemann Zeros

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 N̅ (E) and a “fluctuation.” Berry and Keating have shown that the asymptotic expansion of N̅ (E) counts states of positive energy less than E in a “regularized” semiclassical 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).