RANDOM-MATRIX THEORY AND THE RIEMANN ZEROS .2. N-POINT CORRELATIONS

Montgomery has conjectured that the non-trivial zeros of the Riemann z eta-function are pairwise distributed like the eigenvalues of matrices in the Gaussian unitary ensemble (GUE) of random matrix theory (RMT). In this respect, they provide an important model for the statistical properties of the energy levels of quantum systems whose classical lim its are strongly chaotic. We generalize this connection by showing tha t for all n greater than or equal to 2 the n-point correlation functio n of the zeros is equivalent to the corresponding GUE result in the ap propriate asymptotic limit. Our approach is based on previous demonstr ations for the particular cases n = 2, 3, 4. It relies on several new combinatorial techniques, first for evaluating the multiple prime sums involved using a Hardy-Littlewood prime-correlation conjecture, and s econd for expanding the GUE correlation-function determinant. This con stitutes the first complete demonstration of RMT behaviour for all ord ers of correlation in a simple, deterministic model.