Pauli graphs, Riemann hypothesis, and Goldbach pairs

We consider the Pauli group $\mathcal{P}_q$ generated by unitary quantum generators $X$ (shift) and $Z$ (clock) acting on vectors of the $q$-dimensional Hilbert space. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)$ and that there exists a specific inequality involving the Euler constant $\gamma\sim0.577$ 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 $p<n-1$ with $p$ not dividing $n$ and such that $n-p$ is prime. In the extreme high-dimensional case, at primorial numbers $N_r$, the Hardy–Littlewood function $R(q)$ is introduced for estimating the number of Goldbach pairs, and a new inequality (Theorem $4$) is established for the equivalence to the Riemann hypothesis in terms of $R(N_r)$. We discuss these number-theoretical properties in the context of the qudit commutation structure.