Riemann hypothesis and Quantum Mechanics

In their 1995 paper, Jean-Benôıt Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function ζ(β), where β is an inverse temperature. We formulate Riemann hypothesis (RH) as a property of the low temperature Kubo-Martin-Schwinger (KMS) states of this theory. More precisely, the expectation value of the BC phase operator can be written as φβ(q) = N β−1 q−1 ψβ−1(Nq), where Nq = ∏q k=1 pk is the primorial number of order q and ψb a generalized Dedekind ψ function depending on one real parameter b as ψb(q) = q ∏ p∈P,p|q 1− 1/p 1− 1/p . Fix a large inverse temperature β > 2. The Riemann hypothesis is then shown to be equivalent to the inequality Nq|φβ(Nq)|ζ(β − 1) > e log logNq, for q large enough. Under RH, extra formulas for high temperatures KMS states (1.5 < β < 2) are derived.