# Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

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 ?

Thank you.

• I remember having seen a talk of Konsevitch of that matter at IHES, I think he even wrote a paper/slides about that, you could try to find that maybe. – ujsgeyrr1f0d0d0r0h1h0j0j_juj Mar 2 '15 at 17:22
• To the best of my knowledge, no, there is no such statement on the random matrix theory side that would imply the Riemann hypothesis. The connection between $\zeta(s)$ and random matrix theory is very conjectural - so much so that one essentially has to assume RH first before trying to say anything about the random matrix theoretic properties of the zeroes of $\zeta(s)$. – Peter Humphries Mar 2 '15 at 17:59
• No; the main thing is that random matrices can give, at best, "measure zero" type results. Right now we know that at least 44% of the zeroes of the Riemann zeta function are simple and lie on the critical line. The absolute best random matrices could ever say is 100%, meaning still the possibility of a finite number of counterexamples, or a thin infinite set of counterexamples of RH. – Will Jagy Mar 2 '15 at 19:21