Montgomery pair correlation conjecture

The conjecture that, at large heights, normalized pairwise spacings of nontrivial zeros of the Riemann zeta function follow the pair-correlation law of eigenvalues in the Gaussian unitary ensemble.

Pair correlation function: A function describing the statistical distribution of pairs of points as a function of their separation. The conjecture asserts a specific limiting pair-correlation function for normalized zeta zeros.
Riemann zeta function: The meromorphic function defined initially by ζ(s)=Σ n⁻ˢ for Re(s)>1 and continued to the complex plane. Its nontrivial zeros supply the points whose pair spacings are studied.
Hugh Montgomery: An American mathematician whose work on analytic number theory includes the pair-correlation formula for zeta zeros. Montgomery derived the restricted pair-correlation result that bears his name.
Gaussian unitary ensemble: An ensemble of complex Hermitian random matrices whose probability distribution is invariant under unitary conjugation. Its bulk eigenvalue pairs have the correlation law predicted for normalized zeta zeros.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one-half. It is unproved, and its truth is commonly assumed in formulations concerning zeros on the critical line.
Montgomery's pair correlation formula: An asymptotic formula for correlations between pairs of zeta zeros, proved by Hugh Montgomery under a restriction on test functions. Its restricted-range result is the theorem underlying the conjecture.
Nontrivial zeros of the Riemann zeta function: The zeros of ζ(s) in the critical strip 0<Re(s)<1. Their ordinates form the sequence whose local pair statistics the conjecture predicts.
Andrew Odlyzko: An American mathematician known for large-scale computations of zeros of the Riemann zeta function. Odlyzko's numerical data at high zero indices strongly supported the predicted spacing statistics.
GUE hypothesis: The conjecture that suitably scaled zeta-zero statistics agree with those of the Gaussian unitary ensemble. This broader statistical prediction includes the pair-correlation claim as one of its central consequences.
Pair correlation conjecture for L-functions: A family of conjectures predicting pair statistics for zeros of L-functions from random-matrix ensembles. Generalizing the zeta prediction may reveal which arithmetic features determine the relevant statistics.

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