RANDOM-MATRIX THEORY AND THE RIEMANN ZEROS .1. 3-POINT AND 4-POINT CORRELATIONS

The non-trivial zeros of the Riemann zeta-function have been conjectur ed to be pairwise distributed like the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) of random matrix theory. They therefor e behave like the energy levels of quantum systems whose classical lim it is strongly chaotic and non-invariant with respect to time-reversal . We show that this analogy extends directly to higher-order statistic s. Starting with an explicit formula relating the zeros to the prime n umbers (the analogue of the Gutzwiller trace formula of quantum chaolo gy), we demonstrate that the 3-point and 4-point zero correlation func tions 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 st udy of 2-point correlations and involves the introduction of several n ew techniques. These will form the basis of a demonstration, to be des cribed in another paper, that the equivalence extends to the general n -point correlation function.