Speaker
Description
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).
