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 $\overline{N}(E)$ and a “fluctuation.” Berry and Keating have shown that the asymptotic expansion of $\overline{N}(E)$ 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 $N(E)$.

Figure 1

Absolute value of the wave function ${\psi }_E^{+}(x,y)$ for $E=10$ in the interval $-10<x,y<10$. The picture on the left gives a 3D representation while that on the right is a density plot.

Figure 2

Left: $xy$ plane with the classical trajectories $xy=E{\ell }^2$. Right: sphere that results from the identifications of the edges. Now the open trajectories become closed trajectories in one of four segments, enclosing one of the equatorial points $A$, $B$, $C$, $D$.