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Montgomery Pair Correlation

The conjecture that normalized spacings of Riemann zeta zeros match GUE eigenvalue statistics.

In 1972 Hugh Montgomery conjectured that the pair correlation function of the nontrivial zeros of ζ(s), suitably normalized, reproduces the pair correlation of eigenvalues of a large random Hermitian matrix drawn from the Gaussian Unitary Ensemble.

The correlation kernel is R2(α) = 1 − (sin πα / πα)². It has been verified numerically to twenty billion zeros and analytically for restricted test functions.

For neutronics this identity is operational, not aesthetic: it lets us borrow the algebraic machinery of random matrix theory to close reactor spectral problems in analytical form.

// related concepts

GUE Random Matrix Theory

The Gaussian Unitary Ensemble — Hermitian matrices whose eigenvalue statistics govern chaotic quantum systems.

Nontrivial Riemann Zeros

The complex zeros of ζ(s) lying on the critical line Re(s) = ½ — the skeleton of the spectral operator.

Spectral Neutronics

Closed-form eigenexpansion of the neutron transport operator over a Riemann-anchored basis.