LMI Seminar: Factorization of numbers, Schrödinger cats and the Riemann hypothesis
Wolfgang P. Schleich
In this talk we connect the three different topics of factorization of numbers, Schrödinger cats and the Riemann hypothesis. The bridge between these areas is the concept of a Gauss sum.
Gauss sums manifest themselves in various phenomena such as the Talbot effect, wave packet dynamics or quantum carpets. Moreover, Gauss sums can be used to efficiently factor numbers. The talk summarizes these activities  and discusses a new approach  based on a potential with a logarithmic energy spectrum.
Moreover, we propose an elementary quantum system which provides us with the Riemann zeta function. We show  that its zeroes are a consequence of the interference of two quantum systems with opposite phases. However, the preparation of such a superposition state (Schrödinger cat) is impossible unless one takes advantage of entangled quantum systems. In this sense analytic continuation familiar from complex analysis finds entanglement as its analogue in quantum mechanics.
We conclude by introducing a geometrical approach  towards the Riemann hypothesis based on the lines of constant phase.
 S. Wölk, W. Merkel, W.P. Schleich, I.Sh. Averbukh, B. Girard, and G.Paulus, Factorization of numbers with Gauss sums: I. Mathematical background, New J. Phys. 13, 103007 (2011)
 F. Gleisberg, F. Di Pumpo, G. Wolff, and W.P. Schleich, Prime factorization of arbitrary integers with a logarithmic energy spectrum, J. Phys. B: At. Mol. Opt. Phys. 51, 035009 (2018)
 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)
 W.P. Schleich, I. Bezdekova, M.B. Kim, P.C. Abbott, H. Maier, H.L. Montgomery, and J.W. Neuberger, Equivalent formulations of the Riemann hypothesis based on lines of constant phase, Phys. Scr. 93, 065201 (2018)