Pauli graphs, Riemann hypothesis, Goldbach pairs

Let consider the Pauli group ${\mathcal{P}}_q=<X,Z>$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X\mathrm{\mid }s>=\mathrm{\mid }s+1>$ and $Z\mathrm{\mid }s>={\omega }^s\mathrm{\mid }s>$, with $\omega =\exp (2i\pi \mathrm{/}q)$. It has been found that the number of maximal mutually commuting sets within ${\mathcal{P}}_q$ is controlled by the Dedekind psi function $\psi (q)=q{\prod }_{p\mathrm{\mid }q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$,involvingtheEulerconstant$\gamma \sim 0.577$,thatisonlysatisfiedatspecificlowdimensions$q \in \mathcal {A}={2,3,4,5,6,8,10,12,18,30}$\mathrm{.}Theset$$\mathcal{A}$$ 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.