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<h1 align=center> Dynamical system problem studied by Bost--Connes
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We state the problem solved by Bost and Connes in [BC] (J-B.~Bost,
A.~Connes, Selecta Math. (New Series), {\bf 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 {\bf 2} (1997) 115--138) and [Coh](Paula B Cohen, J.
Th\'eorie des Nombres de Bordeaux, {\bf 11} (1999), 15--30). A
(unital) $C^*$-algebra $B$ is an (unital) algebra over the
complex numbers ${\mathbb C}$ with an adjoint $x\mapsto x^*$, $x\in B$,
that is, an anti-linear map with $x^{**}=x$, $(xy)^*=y^*x^*$,
$x,y\in B$, and a norm $\|\,.\,\|$ with respect to which $B$ is
complete and addition and multiplication are continuous
operations. One requires in addition that $\|xx^*\|=\|x\|^2$ for
all $x\in B$. A $C^\ast$ dynamical system is a pair
$(B,\sigma_t)$, where $\sigma_t$ is a 1-parameter group of
$C^*$-automorphisms $\sigma:{\mathbb R}\mapsto{\rm Aut}(B)$. A state
$\varphi$ on a $C^*$-algebra $B$ is a positive linear functional
on $B$ satisfying $\varphi(1)=1$. The definition of
Kubo-Martin-Schwinger (KMS) of an equilibrium state at inverse
temperature $\beta$ is as follows.

{\bf Definition}: {\it Let $(B,\sigma_t)$ be a dynamical system,
and $\varphi$ a state on $B$. Then $\varphi$ is an equilibrium
state at inverse temperature $\beta$, or ${\rm KMS}_\beta$-state, if for each
$x,y\in B$ there is a function $F_{x,y}(z)$, bounded and
holomorphic in the band $0<{\rm Im}(z)<\beta$ and continuous on its
closure, such that for all $t\in {\mathbb R}$,
\begin{equation}
F_{x,y}(t)=\varphi(x\sigma_t(y)),\qquad
F_{x,y}(t+\sqrt{-1}\beta)=\varphi(\sigma_t(y)x).
\end{equation}}

A symmetry group $G$ of the dynamical system $(B,\sigma_t)$ is a
subgroup of ${\rm Aut}(B)$ commuting with $\sigma$:
$$
g\circ\sigma_t=\sigma_t\circ g,\qquad g\in G, t\in {\mathbb R}.$$ Consider
now a system $(B,\sigma_t)$ with interaction. Then, guided by
quantum statistical mechanics, we expect that at a critical
temperature $\beta_0$ a phase transition occurs and the symmetry is
broken. The symmetry group $G$ then permutes transitively a
family of extremal ${\rm KMS}_\beta$- states generating the possible
states of the system after phase transition: the ${\rm KMS}_\beta$-state
is no longer unique. This phase transition phenomenon is known as
spontaneous symmetry breaking at the critical inverse temperature
$\beta_0$. We state the problem related to the \begin{rawhtml}<a href="../4a">Riemann zeta function</a>\end{rawhtml} and solved by Bost and Connes. 

\noindent{\bf Problem 1:} {\it Construct a dynamical system
$(B,\sigma_t)$ with partition function the zeta function $\zeta(\beta)$
of Riemann, where $\beta>0$ is the inverse temperature, having
spontaneous symmetry breaking at the pole $\beta=1$ 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 $W=\prod_p {\mathbb Z}_p^*$ where the product is over the primes
$p$ and ${\mathbb Z}_p^*=\{u_p\in{\mathbb Q}_p: {|u_p|}_p=1\}$. We use here the
normalisation $|p|_p=p^{-1}$. This is the same as the Galois
group ${\rm Gal}({\mathbb Q}^{ab}/{\mathbb Q})$, where ${\mathbb Q}^{ab}$ is the maximal
abelian extension of the rational number field ${\mathbb Q}$. 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 ${\mathbb Q}^*$ into
the completions ${\mathbb Q}_p$ of ${\mathbb Q}$ with respect to the prime
valuations $|.|_p$. The following natural generalisation of this problem to
the number field case and the \begin{rawhtml}<a href="../66a">Dedekind zeta function</a>\end{rawhtml} was solved in [Coh] (see also [HL], [ALR]).

\noindent{\bf Problem 2:} {\it Given a number field $K$,
construct a dynamical system $(B,\sigma_t)$ with partition
function the Dedekind zeta function $\zeta_K(\beta)$, where $\beta>0$ is
the inverse temperature, having spontaneous symmetry breaking at
the pole $\beta=1$ of the Dedekind function with respect to a natural
symmetry group.}

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<p> <br>
Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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