Riemann hypothesis and Quantum Mechanics

In their 1995 paper, Jean-Benoît 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 $ϕ_β(q)=N_{q-1}^{β-1} ψ_{β-1}(N_q),$ where $N_q=\prod_{k=1}^qp_k$ is the primorial number of order $q$ and $ψ_b$ a generalized Dedekind $ψ$ function depending on one real parameter $b$ as $ψ_b (q)=q \prod_{p \in \mathcal{P,}p \vert q}\frac{1-1/p^b}{1-1/p}.$ Fix a large inverse temperature $β>2.$ The Riemann hypothesis is then shown to be equivalent 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.

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