  The quantum weak value draws many attentions recently from theoretical
curiosity to experimental applications. Now we design an unusual weak measuring
procedure as the pre-selection, mid-selection and post-selection to study the
correlation function of two weak values, which we called the weak correlation
function. In this paper, we proposed an weak measurement experiment to measure
the canonical commutator $[\hat{x},\hat{p}]=i\hbar$ in quantum mechanics.
Furthurmore, we found the intriguing equivalence between the canonical
commutation relation and Riemann hypothesis, and then obtained the weak value
of nontrivial Riemann zeros. Finally, as an nontrivial example of weak
correlations, we also passed successfully a testing on the (anti-)commutators
of Pauli operators, which followed the experimental setup of the landmark paper
of Aharonov, et al. in 1988. Our proposed experiments could hopefully test the
fundamental canonical relationship in quantum worlds and trigger more testing
experiments on weak correlations.
