THE N-LEVEL CORRELATIONS OF ZEROS OF THE ZETA-FUNCTION

We define the n-level correlation sums for the normalized zeros of Rie mann's zeta function. A special case of these is the pair correlation function investigated by H. Montgomery. If one assumes the Riemann Hyp othesis, then these sums measure the n-level correlations of the imagi nary parts of the zeros, a statistic introduced by F. Dyson. It is sho wn that for a restricted class of test functions the n-level correlati ons follow the distribution predicted by the Gaussian Unitary Ensemble of random matrix theory. More generally, the same is true universally for the principal L-functions attached to cuspidal automorphic repres entations of GL(m) over the rationals.