J. Phys. IV France 08 (1998) Pr6-189-Pr6-195
Riemann hypothesis and dynamical systemsA. Bonelli1 and M. Rasetti2
1 Dipartimento di Chimica, Università della Basilicata, via Nazario Sauro 85, 85100 Potenza, Italy
2 Dipartimento di Fisica and Unità INFM, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy
The possible connection of Riemann's Hypothesis on the non trivial zeroes of the zeta function ζ(z) with the theory of dynamical systems, both quantum and classical, is discussed. The conjecture of the existence of an underlying integrable structure is analysed, resorting on the one hand to the link between Riemann's zeta function and the Selberg trace formula, on the other to the relation between the zeroes of ζ(z) and the Gauss unitary ensemble of random matrices, to which - through basic results on the twisted de Rham cohomology - a holonomic system of completely integrable differential equations can be associated.
© EDP Sciences 1998