Riemann hypothesis and quantum mechanics.

In their 1995 paper, Jean-Benoit Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function z(b), where b 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 \phi _{\beta }(q)=N_{q-1}^{\beta -1} \psi _{\beta -1}(N_q), where Nq = [?]qk = 1pk is the primorial number of order q and psb is a generalized Dedekind ps function depending on one real parameter b as \psi _b (q)=q \prod _{p \in \mathcal {P,}p \vert q}\frac{1-1/p^b}{1-1/p}. Fix a large inverse temperature b > 2. The RH is then shown to be equivalent to the inequality N_q |\phi _\beta (N_q)|\zeta (\beta -1) \gt {\rm e}^\gamma \log \log N_q, for q large enough. Under RH, extra formulas for high-temperature KMS states (1.5 < b < 2) are derived.'Number theory is not pure Mathematics. It is the Physics of the world of Numbers.'Alf van der Poorten [ABSTRACT FROM AUTHOR]