  Let $\mathbf{F}_q$ be a finite field of $q$ elements. For multiplicative
characters $\chi_1,\dots, \chi_m$ of $\mathbf{F}_q^\times$, we let
$J(\chi_1,\dots, \chi_m)$ 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 $q\to \infty$, when $\chi_1$ and $\chi_2$
run through all nontrivial multiplicative characters of $\mathbf{F}_q^\times$.
In this paper, we show a similar property for $m\ge 2$. More generally, we show
that the normalized Jacobi sum $q^{-(m-1)/2}J(\chi_1,\dots,\chi_m)$
($\chi_1\dotsm \chi_m$ nontrivial) is asymptotically equidistributed on the
unit circle, when $\chi_1,\dots, \chi_m$ run through arbitrary sets of
nontrivial multiplicative characters of $\mathbf{F}_q^\times$ with two of the
sets being sufficiently large. The case $m=2$ answers a question of
Shparlinski.
