Hecke algebras,type III factors and phase transitions with spontaneous symmetry breaking in number theory

In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings provides the desiredC*-dynamical system, which admits the function as partition function and the Galois group Gal(^cycl/) of the cyclotomic extension ^cycl of as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature =1 (cf. Bos-C]). The original motivation for these results comes from the work of B. Julia J] (cf. also Spe]).