Montgomery pair correlation conjecture

Establish that the pair correlation of normalized ordinates of nontrivial zeros of the Riemann zeta function equals the Gaussian Unitary Ensemble pair-correlation function in the stated scaling limit.

Background

Montgomery formulated a precise conjecture for the two-point correlation of zeta zeros after rescaling by the local mean spacing, predicting agreement with GUE statistics.

This conjecture initiated deep connections between number theory and random matrix theory and has strong numerical support (e.g., Odlyzko’s computations), but no proof.

References

In 1973, Hugh Montgomery conjectured a striking statistical property of the nontrivial zeros of the Riemann zeta function on the critical line. Specifically, he conjectured that for $0 < a < b$, and $N(T)=\sum_{0<\gamma \leq T} 1$, $$ \lim_{T \to \infty} \frac{1}{N(T)} # \left{ (\gamma, \gamma’) : 0 < \gamma, \gamma’ < T,\; \frac{2\pi a}{\log T} \leq |\gamma - \gamma’| \leq \frac{2\pi b}{\log T} \right} = \int_ab \left( 1 - \left( \frac{ \pi u}{\pi u} \right)2 \right) du, $$

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Subsubsection Montgomery’s pair correlation