  We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about
the $n$-level correlations of eigenvalues of random unitary matrices as well as
a new proof of a formula due to M. Diaconis and P. Shahshahani expressing
averages of trace products over the unitary matrix ensemble. Our method uses
the zero statistics of Artin-Schreier L-functions and a deep equidistribution
result due to N. Katz and P. Sarnak.
