On the distribution of Jacobi sums

Let F-q be a finite field of q elements. For multiplicative characters chi(1),..., chi(m) of F-q(x), we let J (chi(1), ..., chi(m)) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m = 2, the normalized Jacobi sum q(-1/2) J (chi(1) , chi(2)) (chi(1), chi(2) nontrivial) is asymptotically equidistributed on the unit circle as q -> infinity, when chi(1) and chi(2) run through all nontrivial multiplicative characters of F-q(x). In this paper, we show a similar property for m >= 2. More generally, we show that the normalized Jacobi sum q(-(m-1)/2) J (chi(1), ..., chi(m)) (chi(1) ... chi(m) nontrivial) is asymptotically equidistributed on the unit circle, when chi(1), ..., chi(m) run through arbitrary sets of nontrivial multiplicative characters of F-q(x) with two of the sets being sufficiently large. The case m = 2 answers a question of Shparlinski.