4 pages, 2 figures.-- PACS numbers: 02.10.De, 05.45.Mt.-- ArXiv pre-print available at: http://arxiv.org/abs/0805.4079
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 Ñ(E) and a 'fluctuation'. Berry and Keating have shown that the asymptotic expansion of Ñ(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).
This work was supported by the CICYT
project FIS2004-04885 (G.S.) and by the EPSRC (P.K.T.). GS also acknowledges ESF Science Programme INSTANS 2005-2010.