Number theory, dynamical systems and statistical mechanics

We shortly review recent work interpreting the quotient ζ(s− 1)/ζ(s) of Riemann zeta functions as a dynamical zeta function. The corresponding interaction function (Fourier transform of the energy) has been shown to be ferromagnetic, i.e. positive. On the additive group Gk := (Z/2Z), with Z/2Z = ({0, 1},+). we set inductively h0 := 1, hk+1(σ, 0) := hk(σ) and hk+1(σ, 1) := hk(σ) + hk(1− σ), (1) where σ = (σ1, . . . , σk) ∈ Gk and 1− σ := (1− σ1, . . . , 1− σk) is the inverted configuration. The sequences hk(σ) of integers, written in lexicographic order, coincide with the denominators of the modified Farey sequence. We now formally interpret σ ∈ Gk as a configuration of a spin chain with k spins and energy function Hk := ln(hk). Thus we may interpret