Riemann hypothesis and Quantum Mechanics

Abstract

In their 1995 paper, Jean-Benoît Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function $\zeta (\beta )$, where $\beta$ 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 $${\varphi }_{\beta }(q)=N_{q-1}^{\beta -1}{\psi }_{\beta -1}(N_q),$$ where $N_q={\prod }_{k=1}^q{p}_k$ is the primorial number of order $q$ and ${\psi }_b$ a generalized Dedekind $\psi$ function depending on one real parameter $b$ as $${\psi }_b(q)=q\prod _{p\in \mathcal{P},p|q}\frac{1-1/p^b}{1-1/p}.$$ Fix a large inverse temperature $\beta >2.$ The Riemann hypothesis is then shown to be equivalent to the inequality $$N_q|{\varphi }_{\beta }(N_q)|\zeta (\beta -1)>e^{\gamma }\log \log N_q,$$ for $q$ large enough. Under RH, extra formulas for high temperatures KMS states ($1.5<\beta <2$) are derived.

version to appear in J. Phys. A: Math. Theor