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.