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Riemann hypothesis and quantum mechanics

2011; Institute of Physics; Volume: 44; Issue: 14 Linguagem: Inglês

10.1088/1751-8113/44/14/145203

ISSN

1751-8121

Autores

Michel Planat, Patrick Solé, Sami Omar,

Tópico(s)

Analytic Number Theory Research

Resumo

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 where Nq = ∏qk = 1pk is the primorial number of order q and ψb is a generalized Dedekind ψ function depending on one real parameter b as Fix a large inverse temperature β > 2. The RH is then shown to be equivalent to the inequality for q large enough. Under RH, extra formulas for high-temperature KMS states (1.5 < β < 2) are derived.

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