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

We note in passing that a profound connection to number theory arises here through the work of Montgomery , who conjectured that the pair correlation of the non-trivial zeros of the Riemann zeta function follows the GUE (Gaussian Unitary Ensemble) model of Random Matrix Theory.

— On the asymptotic behavior of the number variance of random points on the unit torus  (2610.11687 - Aistleitner et al., 8 Oct 2026) in Section 1, Introduction and statement of results

Goldston, Lee, Schettler, and Suriajaya [10] showed that the full pair correlation conjecture, without separately assuming the Riemann hypothesis, implies that asymptotically almost all zeros are simple and lie on the critical line.

— More than 8/9 zeros of the Riemann zeta function are simple or on the critical line  (2610.23467 - Cobeli et al., 16 Oct 2026) in Section 1, Introduction, page 2

The first follows from Montgomery's pair correlation conjecture (PCC) which asserts that assuming RH we have \begin{equation}\label{Eq:PairCorrelationConj} \frac{1}{N(T)}#\left{0<\gamma_n, \gamma_m T : 0< \gamma_n-\gamma_m \frac{2\pi \alpha}{\log T}\right}= {P}(\alpha)+o(1), \end{equation} for any fixed $\alpha 0$, where \begin{equation}\label{Eq:DefinitionPAlpha} {P}(\alpha):= \int_0{\alpha} \left(1-\left(\frac{\sin \pi u}{\pi u}\right)2\right) du. \end{equation}

Eq:PairCorrelationConj:

1N(T)#{0<γn,γmT:0<γn−γm2παlog⁡T}=P(α)+o(1),\frac{1}{N(T)}\#\left\{0<\gamma_n, \gamma_m T : 0< \gamma_n-\gamma_m \frac{2\pi \alpha}{\log T}\right\}= {P}(\alpha)+o(1),

Eq:DefinitionPAlpha:

P(α):=∫0α(1−(sin⁡πuπu)2)du.{P}(\alpha):= \int_0^{\alpha} \left(1-\left(\frac{\sin \pi u}{\pi u}\right)^2\right) du.

— Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet $L$-functions  (2610.26841 - Lamzouri, 19 Oct 2026) in Section 1, Introduction