  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 $X|s> =|s+1>$ and $Z|s> =\omega^s |s>$, 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.
