21734 articles – 15570 Notices  [english version]
 HAL : hal-00576269, version 1
 arXiv : 1103.2608
 Theoretical and Mathematical Physics 171, 3 (2012) 780-791
 Pauli graphs, Riemann hypothesis, Goldbach pairs
 Michel Planat 1, Fabio Anselmi 1
 (01/06/2012)
 Let consider the Pauli group $\mathcal{P}_q=\left\langle X,Z\right\rangle$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X\left|s\right \rangle=\left|s+1\right \rangle$ and $Z\left|s\right \rangle=\omega^s \left|s\right \rangle$, with $\omega=\exp(2i\pi/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$, involving the Euler constant $\gamma \sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q }$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^{\gamma}$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.
 1 : Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (FEMTO-ST) CNRS : UMR6174 – Université de Franche-Comté – Université de Technologie de Belfort-Montbeliard – Ecole Nationale Supérieure de Mécanique et des Microtechniques 2 : Télécom ParisTech Institut Mines-Télécom
 Domaine : Mathématiques/Physique mathématiquePhysique/Physique QuantiqueMathématiques/Théorie des nombres
 Mots Clés : Pauli graphs – number theory
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 hal-00576269, version 1 http://hal.archives-ouvertes.fr/hal-00576269 oai:hal.archives-ouvertes.fr:hal-00576269 Contributeur : Michel Planat <> Soumis le : Lundi 14 Mars 2011, 08:58:32 Dernière modification le : Mardi 14 Août 2012, 10:22:30