Random matrix theory and the Riemann zeros .2. N-point correlations

Montgomery has conjectured that the non-trivial zeros of the Riemann zeta-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 limits are strongly chaotic. We generalize this connection by showing that for all n greater than or equal to 2 the n-point correlation function of the zeros is equivalent to the corresponding GUE result in the appropriate asymptotic limit. Our approach is based on previous demonstrations 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 second for expanding the GUE correlation-function determinant. This constitutes the first complete demonstration of RMT behaviour for all orders of correlation in a simple, deterministic model.