On a ferromagnetic spin chain (Q1803545)

As confessed by the author himself, the aim of the paper is to relate ideas and concepts from statistical mechanics to the Riemann zeta- function. Specifically, he manages to prove that the quotient \(Z(s)=\zeta(1-s)/\zeta(s)\) of Riemann zeta functions is the partition function of an infinite ferromagnetic spin chain for inverse temperature \(s\). The existence of a connection between number theory and statistical mechanics was already conjectured by Kac many years ago, mainly motivated by the Lee-Yang circle theorem of statistical mechanics, which states that all zeros of the partition function of a ferromagnetic Ising model have unit modulus when represented in the complex activity plane. So the author becomes interested in the zeros of \(Z(s)\) in the complex \(s\)- plane, \(s\) being the inverse temperature. The usual simplifications of the Lee-Yang circle theorem, which apply to certain ferromagnets and predict zero-free half-planes of the inverse temperature, do not apply in this situation, because the spin chain includes many-body interactions. The author starts by (uniformly) approximating \(Z(s)\) by the partition function \(Z_ k\) of a chain of \(k\) classical spins. There are no couplings to an external field and the interaction coefficients between an odd number of spins vanish. The canonical and grandcanonical ensembles are shown to be (weakly) ferromagnetic, and inequalities between the interaction coefficients for spin chains of length \(k\) and of length \(k+1\), and also upper bounds for these coefficients are obtained by the author. The results of a numerical computation are provided in an appendix. Finally, a weak form of asymptotic translation invariance and estimates for the decay properties of the model are given.