Pauli graphs, Riemann hypothesis, and Goldbach pairs

We consider the Pauli group 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 P-q is controlled by the Dedekind psi function psi(q) and that there exists a specific inequality involving the Euler constant gamma similar to 0.577 that is only satisfied at specific low dimensions q is an element of A = {2, 3, 4, 5, 6, 8, 10, 12, 18, 30}. The set A is closely related to the set Aa(a){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.