HP Labs Technical Reports

Random Matrix Theory and the Riemann Zeros II: n-Point Correlations

Bogomolny, E.B.; Keating, J.P.

HPL-BRIMS-96-13

Keyword(s): random matrix theory; Riemann zeta function; spectral statistics; quantum chaos

Abstract: Montgomery (1973) has conjectured that the non- trivial zeros of the Riemann zeta-function are pairwise distributed like 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 generalise this connection by showing that for all n is greater 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 (Keating 1993, Bogomolny and Keating 1995). 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.

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