Comment on "Landau Levels and Riemann Zeros"

Abstract

Sierra and Townsend (ST) [1] showed that the ``absorption spectrum'' model of Connes [2] arises as the lowest-Landau-level (LLL) limit of a charged particle in crossed electric and magnetic fields, and suggested that the higher Landau levels supply the fluctuation part of the Riemann-zero counting function $N(E)$. Within the approximation used by ST we construct the eigenfunctions of an arbitrary Landau level $n$ and derive their asymptotics at the two box boundaries that ST identify. The boundary condition is the same as for $n=0$ up to an $E$-independent phase $(-\ii)^n$, so no $\ln E$ term arises for any fixed $n$. We also note that the $\ln E$ term in the semiclassical Eq.~(29) of ST is the first-order shift of the smooth LLL counting function, $E\to E+γz^2$.