arXiv:1012.4665v3 [math-ph] 16 Feb 2011 RIEMANN HYPOTHESIS AND QUANTUM MECHANICS MICHEL PLANAT, PATRICK SOL ´ E, AND SAMI OMAR ”Number theory is not pure Mathematics. It is the Physics of the world of Numbers.” Alf van der Poorten. Abstract. In their 1995 paper, Jean-Benoˆ ıt 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 φ β (q)= N β-1 q-1 ψ β-1 (N q ), where N q =  q k=1 p k is the primorial number of order q and ψ b a generalized Dedekind ψ function depending on one real parameter b as ψ b (q)= q  p∈P,p|q 1 − 1/p b 1 − 1/p . Fix a large inverse temperature β> 2. The Riemann hypothesis is then shown to be equiv- alent to the inequality N q |φ β (N q )|ζ (β − 1) >e γ log log N q , for q large enough. Under RH, extra formulas for high temperatures KMS states (1.5 <β< 2) are derived. 1. Introduction The Riemann Hypothesis (RH), that describes the non trivial zeroes of Riemann ζ func- tion, is a Holy Grail of Mathematics [1, 9]. Many formulations of RH may be found in the literature [5]. In this paper, we mainly refer to the Nicolas inequality [14] (1) N k ϕ(N k ) >e γ log log N k , where N k =  k i=1 p i is the primorial of order k, ϕ is the Euler totient function and γ ≈ 0.577 is the Euler-Mascheroni constant. Key words and phrases. Bost-Connes model, Riemann Hypothesis, Primorial numbers, Quantum Infor- mation. MSC codes: 46L05, 11M26, 11A25, 81P68. 1