  We use an estimate for character sums over finite fields of Katz to solve
open problems of Montgomery and Turan. Let h=>2 be an integer. We prove that
inf_{|z_k| => 1} max_{v=1,...,n^h} |sum_{k=1}^n z_k^v| <= (h-1+o(1)) sqrt n.
This gives the right order of magnitude for the quantity and improves on a
bound of Erdos-Renyi by a factor of the order sqrt log n.
