  We introduce a new number-theoretic spin chain and explore its thermodynamics
and connections with number theory. The energy of each spin configuration is
defined in a translation-invariant manner in terms of the Farey fractions, and
is also expressed using Pauli matrices. We prove that the free energy exists
and exhibits a unique phase transition at inverse temperature beta = 2. The
free energy is the same as that of a related, non translation-invariant
number-theoretic spin chain. Using a number-theoretic argument, the
low-temperature (beta > 3) state is shown to be completely magnetized for long
chains. The number of states of energy E = log(n) summed over chain length is
expressed in terms of a restricted divisor problem. We conjecture that its
asymptotic form is (n log n), consistent with the phase transition at beta = 2,
and suggesting a possible connection with the Riemann zeta function. The spin
interaction coefficients include all even many-body terms and are translation
invariant. Computer results indicate that all the interaction coefficients,
except the constant term, are ferromagnetic.
