2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/133245We 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.v1: 9 pages; v2: Minor changes. Fixed error in last three lines of proof of Theorem 2: v3: New title. Minor changesNumber Theory11N30; 11L40text