We consider the Farey Fraction Spin Chain, a set of one-dimensional  statistical mechanical models built on the Farey fractions (modified  Farey sequence).  These models lie between statistical mechanics and  number theory, and are of interest in both areas.  A direct  connection to dynamical systems is used to prove that the models  rigorously exhibit a (barely) second-order phase transition.   Additionally, we calculate certain correlation functions.  One can  determine the phase diagram, including an external magnetic field, by  means of renormalization group and also via a (dynamical system  inspired) cluster approximation.  The results, interestingly, are  almost consistent. Examination of the partition function at the  critical point suggests a subtle, apparently new, property of the  Farey fractions.  There are also rigorous results by number theorists  for the “density of states”  and a close connection to the Lewis  three-term equation, extensively studied in the theory of the Selberg  zeta-function.