nLab Bost-Connes system



Arithmetic geometry


physics, mathematical physics, philosophy of physics

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theory (physics), model (physics)

experiment, measurement, computable physics



The Bost-Connes system (Bost & Connes 1995) is the semigroup crossed product C*-algebra C *(/) ×C^\ast(\mathbb{Q}/\mathbb{Z})\rtimes \mathbb{N}^\times equipped with a canonical 1-parameter flow texp(tH)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:


Further developments:

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

An abstract generalization:

Last revised on September 11, 2023 at 17:13:35. See the history of this page for a list of all contributions to it.