We state the problem solved by Bost and Connes in [BC] (J-B. Bost,
A. Connes, Selecta Math. (New Series), **1**, (1995) 411-457)
and its analogue for number fields, considered in [HL](D.
Harari, E. Leichtnam, Selecta Mathematica, New Series 3 (1997),
205-243), [ALR](J. Arledge, M. Laca, I. Raeburn, Doc.
Mathematica **2** (1997) 115-138) and [Coh](Paula B Cohen, J.
Théorie des Nombres de Bordeaux, **11** (1999), 15-30). A
(unital) -algebra is an (unital) algebra over the
complex numbers with an adjoint , ,
that is, an anti-linear map with , ,
, and a norm with respect to which is
complete and addition and multiplication are continuous
operations. One requires in addition that
for
all . A dynamical system is a pair
, where is a 1-parameter group of
-automorphisms
. A state
on a -algebra is a positive linear functional
on satisfying . The definition of
Kubo-Martin-Schwinger (KMS) of an equilibrium state at inverse
temperature is as follows.

**Definition**: *Let be a dynamical system,
and a state on . Then is an equilibrium
state at inverse temperature , or
-state, if for each
there is a function , bounded and
holomorphic in the band
and continuous on its
closure, such that for all
,
*

(1) |

A symmetry group of the dynamical system is a
subgroup of commuting with :

Consider now a system with interaction. Then, guided by quantum statistical mechanics, we expect that at a critical temperature a phase transition occurs and the symmetry is broken. The symmetry group then permutes transitively a family of extremal - states generating the possible states of the system after phase transition: the -state is no longer unique. This phase transition phenomenon is known as spontaneous symmetry breaking at the critical inverse temperature . We state the problem related to the Riemann zeta function and solved by Bost and Connes.

**Problem 1:** *Construct a dynamical system
with partition function the zeta function
of Riemann, where is the inverse temperature, having
spontaneous symmetry breaking at the pole of the zeta
function with respect to a natural symmetry group.*

The symmetry group turns out to be the unit group of the ideles, given by where the product is over the primes and . We use here the normalisation . This is the same as the Galois group , where is the maximal abelian extension of the rational number field . The interaction detected in the phase transition comes about from the interaction between the primes coming from considering at once all the embeddings of the non-zero rational numbers into the completions of with respect to the prime valuations . The following natural generalisation of this problem to the number field case and the Dedekind zeta function was solved in [Coh] (see also [HL], [ALR]).

**Problem 2:** *Given a number field ,
construct a dynamical system with partition
function the Dedekind zeta function
, where is
the inverse temperature, having spontaneous symmetry breaking at
the pole of the Dedekind function with respect to a natural
symmetry group.*

Back to the
main index
for The Riemann Hypothesis.