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arXiv · 0805.4079

Landau levels and Riemann zeros

German Sierra·Paul K. Townsend
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Abstract

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 $\bar N(E)$ and a `fluctuation'. Berry and Keating have shown that the asymptotic expansion of $\bar N(E)$ 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 $N(E)$.

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German Sierra, Paul K. Townsend. 2008-09-22. Landau levels and Riemann zeros. https://doi.org/10.1103/physrevlett.101.110201

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