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A quantum mechanical model of the Riemann zeros

AuthorsSierra, Germán CSIC ORCID
KeywordsMathematical Physics
High Energy Physics - Theory
Number Theory
Quantum Physics
Issue Date5-Dec-2007
PublisherDeutsche Physikalische Gesellschaft
Institute of Physics Publishing
CitationarXiv:0712.0705v1 [math-ph]
New J. Phys. 10 (2008) 033016
AbstractIn 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wave functions, associated to the Berry-Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wave functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model.
Description42 pages, 12 figures.-- PACS numbers: 02.10.De, 05.45.Mt, 11.10.Hi.-- Published in: New Journal of Physics 10 (2008) 033016, available at: (open-access).
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