nLab Bost-Connes system

Idea

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.

References

The original article is

• Jean-Benoit Bost; Alain Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Mathematica. New Series 1 (3): 411–457, (1995) doi:10.1007/BF01589495, ISSN 1022-1824, MR 1366621 (pdf)

A review is in

• Alain Connes, Matilde Marcolli, section 22 of A walk in the noncommutative garden (arXiv:0601054)

Further developments include

• Marcelo Laca, Sergey Neshveyev, Mak Trifkovic, Bost-Connes systems, Hecke algebras, and induction (arXiv:1010.4766)

An abstract generalization is proposed in

• Matilde Marcolli, Goncalo Tabuada, Bost-Connes systems, categorification, quantum statistical mechanics, and Weil numbers, arxiv/1411.3223