Random Matrix Approaches in Number Theory

The connection between random matrix theory and the Riemann zeta function was established in 1973 when Montgomery, who had conjectured the 2-point correlations of the Riemann zeros, and Dyson, who was interested in similar statistics of the eigenvalues of ensembles of unitary matrices, realized that the formulae they had discovered independently were in fact identical in a natural asymptotic limit. Further attempts at the verification of this coincidence of Riemann zero and eigenvalue statistics were then produced from various fronts: overwhelming numerical evidence was afforded by the mammoth computations of Andrew Odlyzko (1989); the heuristic work of Bogomolny and Keating (1995) pointed towards the agreement of not just the 2-point correlation function, but all the n-point statistics as well; and Rudnick and Sarnak (1996) proved that the Riemann zeros and the eigenvalues of this random matrix ensemble have the same n-point statistics in a restricted range.