  We study the density of the roots of the derivative of the characteristic
polynomial Z(U,z) of an N x N random unitary matrix with distribution given by
Haar measure on the unitary group. Based on previous random matrix theory
models of the Riemann zeta function zeta(s), this is expected to be an accurate
description for the horizontal distribution of the zeros of zeta'(s) to the
right of the critical line. We show that as N --> infinity the fraction of
roots of Z'(U,z) that lie in the region 1-x/(N-1) <= |z| < 1 tends to a limit
function. We derive asymptotic expressions for this function in the limits x
--> infinity and x --> 0 and compare them with numerical experiments.
