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Landau levels and Riemann zeros

AuthorsSierra, Germán ; Townsend, Paul K.
KeywordsMathematical Physics
Mesoscopic Systems and Quantum Hall Effect
High Energy Physics - Theory
Number Theory
Quantum Physics
[PACS] Algebraic structures and number theory
[PACS] Quantum chaos; semiclassical methods
Issue Date12-Sep-2008
PublisherAmerican Physical Society
CitationPhysical Review Letters 101(11): 110201 (2008)
SeriesIFT-UAM/CSIC 08-26
AbstractThe 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).
Description4 pages, 2 figures.-- PACS numbers: 02.10.De, 05.45.Mt.-- ArXiv pre-print available at: http://arxiv.org/abs/0805.4079
Publisher version (URL)http://dx.doi.org/10.1103/PhysRevLett.101.110201
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