• Editors' Suggestion
  • Open Access

Hamiltonian for the Zeros of the Riemann Zeta Function

Carl M. Bender, Dorje C. Brody, and Markus P. Müller
Phys. Rev. Lett. 118, 130201 – Published 30 March 2017

Abstract

A Hamiltonian operator H^ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of H^ is 2xp, which is consistent with the Berry-Keating conjecture. While H^ is not Hermitian in the conventional sense, iH^ is PT symmetric with a broken PT symmetry, thus allowing for the possibility that all eigenvalues of H^ are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that H^ is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.

  • Received 23 September 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.130201

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

General PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Carl M. Bender1, Dorje C. Brody2,3, and Markus P. Müller4,5

  • 1Department of Physics, Washington University, St. Louis, Missouri 63130, USA
  • 2Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, United Kingdom
  • 3Department of Optical Physics and Modern Natural Science, St. Petersburg National Research University of Information Technologies, Mechanics and Optics, St. Petersburg 197101, Russia
  • 4Departments of Applied Mathematics and Philosophy, University of Western Ontario, Middlesex College, London, Ontario N6A 5B7, Canada
  • 5The Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

Article Text

Click to Expand

References

Click to Expand
Issue

Vol. 118, Iss. 13 — 31 March 2017

Reuse & Permissions
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

×

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×