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The Bost-Connes system (Bost-Connes 95) is the semigroup crossed product C*-algebra $C^\ast(\mathbb{Q}/\mathbb{Z})\rtimes \mathbb{N}^\times$ equipped with a canonical 1-parameter flow $t\mapsto\exp(-t H)$, and as such thought of as a quantum mechanical system.
The point is that the partition function of this system is the Riemann zeta function (Bost-Connes 95, theorem 5 (c) (page 6)).
Alain Connes has proposed that via this relation there might be a way to shed insight on the Riemann hypothesis using tools from quantum mechanics and statistical mechanics (Connes-Marcolli 06). See also at Riemann hypothesis and physics.
The original article is
A review is in
Further developments include
An abstract generalization is proposed in
Last revised on November 13, 2014 at 12:26:43. See the history of this page for a list of all contributions to it.