Katz and Sarnak predicted that for a nice family of L-functions defined over the ring of polynomials with coefficients in the the field of q elements the Frobenii would become equidistributed in a compact matrix Lie group as q tends to infinity. This talk will discuss the terms which vanish as q tends to infinity for certain statistics of the Frobenii, and shows that for the family of L-functions attached to the r-th power residue symbols, one can describe these lower order terms using random matrix theory.