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Random matrix theory and the Riemann zeros. I. Three- and four-point correlations

The non-trivial zeros of the Riemann zeta-function have been conjectured to be pairwise distributed like the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) of random matrix theory. They therefore behave like the energy levels of quantum systems whose classical limit is strongly chaotic and non-invariant with respect to time-reversal. We show that this analogy extends directly to higher-order statistics. Starting with an explicit formula relating the zeros to the prime numbers (the analogue of the Gutzwiller trace formula of quantum chaology), we demonstrate that the 3-point and 4-point zero correlation functions are asymptotically equivalent to the corresponding GUE results. Our method centres around a Hardy-Littlewood conjecture concerning the distribution of the primes. The calculation generalises a previous study of 2-point correlations and involves the introduction of several new techniques. These will form the basis of a demonstration, to be described in another paper, that the equivalence extends to the general n-point correlation function.