We all know that Riemann Hypothesis has many equivalent statements.
After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as eigenvalues of some types of random matrix. This helps us to understand ZEROS of Zeta function better, but it is not clear to me if random matrix theory can help us to prove Riemann Hypothesis.
I want to know if there is any equivalent statement of Riemann Hypothesis using Random Matrix . For example, if there is any statement such as: “if we prove that certain types of random matrix has such such properties, then Riemann Hypothesis is proved”.
Can anyone share such a statement ? or can you point some resources on this topic ?
Here is another related question.
Since Random Matrix theory is related to energy level of physics systems. Is there any equivalent statement of Riemann Hypothesis using physics theory ? I understand Alain Connes was trying to prove RH using this approach, also Hilbert-Polya conjecture is related to this. Is there a statement such as: “if we prove a physics system (for example, a special xyz system) has such such properties, then Riemann Hypothesis is proved”
Can anyone share such a statement ?