Dresden 2026 – scientific programme
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DY: Fachverband Dynamik und Statistische Physik
DY 59: Quantum Chaos and Coherent Dynamics (joint session DY/TT)
DY 59.5: Talk
Friday, March 13, 2026, 10:30–10:45, HÜL/S186
Dynamic origin of quantum chaos signatures in the zeros of the Riemann zeta function by means of periodic orbit theory — •Andreas Hötzinger, Sebastian Hörhold, Juan Diego Urbina, and Klaus Richter — Institut für Theoretische Physik, Universität Regensburg, Germany
In the 90’s, Berry and Keating [1] provided a qualitative, semiclassical analogy to the counting function of the nontrivial Riemann zeros, i.e. the zeros of the famous zeta function (ZF) ζ(s). Similar to Gutzwiller’s trace formula they obtain a result in which the primes play the role of periodic orbits and argue that the so-called Riemann dynamics, underlying the primes, should be chaotic. It is speculated that this system is the classical limit of a Hermitian quantum Hamiltonian which has eigenvalues coinciding with the nontrivial zeros of ζ(1/2 + i tn).
Recently, a promising candidate for such a Hamiltonian has been proposed [2], which has the potential to advance research toward a proof of the Riemann hypothesis. Based on these results, we use a related and simpler, yet non-Hermitian Hamiltonian and consider its semiclassical regime by employing methods from periodic orbit theory.
In this talk, we present our progress in the study of the classical limit of this operator and its dynamics in a complexified phase space. Through this, we hope to unveil a deeper relation between quantum chaos signatures of number theory encoded in the ZF with classical phase space structures.
[1] M. V. Berry and J. P. Keating, SIAM Review 41.2 pp. 236-266
[2] E. Yakaboylu, arXiv:2408.15135
Keywords: periodic orbit theory; semiclassics; Riemann hypothesis; complexified classical dynamics