The number-theoretical spin chain has exactly one phase transition, which is located at inverse temperature fi cr = 2. There the magnetization jumps from one to zero. The energy density, being zero in the low temperature phase, grows at least linearly in fi cr \Gamma fi. 1 Introduction In  one of us showed that the quotient Z(fi) := i(fi \Gamma 1) i(fi) of Riemann zeta functions could be interpreted for fi ? 2 as the canonical partition function of an infinite ferromagnetic spin chain. This partition function was shown to be the thermodynamic limit Z(fi) = lim k!1 Z k (fi) Department of Mathematics, Kerchof Hall, University of Virginia, Charlottesville Va 22903. e-mail: email@example.com y Dipartimento di Fisica, Gruppi Cam e Gtc, Universit`a di Roma "La Sapienza", I-00185 Roma, Italy. e-mail: firstname.lastname@example.org z Technische Universitat, Fachbereich 3 - Mathematik, MA 7--2, Strasse des 17. Juni 135, D-10623 Berlin, Germany. e-mail: knauf@math..