The Riemann Zeta Function and Quantum Mechanics

by Prof. Dr. Wolfgang P. Schleich (Institut für Quantenphysik, Center for Integrated Quantum Science and Technology, Universität Ulm)

Thursday, 28 October 2021 from 10:00 to 11:00 (Europe/Budapest)
at KFKI campus Bldg. 1 ( Meeting Room )
Budapest, Konkoly-Thege Miklós út 29-33.

Wearing a mask is mandatory.
The event will be streamed online on Zoom:

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Meeting ID: 966 6394 8362
Passcode: 780159


The Riemann zeta function ζ plays a crucial role in number theory as well as physics. Indeed, the distribution of primes is intimately connected to the non-trivial zeros of this function. We briefly summarize the essential properties of the Riemann zeta function and then present a quantum mechanical system which when measured appropriately yields ζ. We emphasize that for the representation in terms of a Dirichlet series interference [1] suffices to obtain ζ. However, in order to create ζ along the critical line where the non-trivial zeros are located we need two entangled quantum systems [2]. In this way entanglement may be considered the quantum analogue of the analytical continuation of complex analysis. We also analyze the Newton flows [3, 4] of ζ as well as of the closely related function ξ. Both provide additional insight [5] into the Riemann hypothesis.

[1] R. Mack, J. P. Dahl, H. Moya-Cessa, W.T. Strunz, R. Walser, and W. P. Schleich, Riemann ζ-function from wave packet dynamics, Phys. Rev. A. 82, 032119 (2010).
[2] C. Feiler and W.P. Schleich, Entanglement and analytical continuation: an intimate relation told by the Riemann zeta function, New J. Phys. 15, 063009 (2013).
[3] J. Neuberger, C. Feiler, H. Maier, and W.P. Schleich, Newton flow of the Riemann zeta function: Separatrices control the appearance of zeros, New J. Phys. 16, 103023 (2014).
[4] J.W. Neuberger, C. Feiler, H. Maier, and W.P. Schleich, The Riemann hypothesis illuminated by the Newton flow of ζ, Phys. Scr. 90, 108015 (2015).
[5] W.P. Schleich, I. Bezděková, M.B. Kim, P.C. Abbott, H. Maier, H. Montgomery, and J.W. Neuberger, Equivalent formulations of the Riemann hypothesis based on lines of constant phase, Phys. Scr. 93, 065201 (2018).