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(vertical bar zk vertical bar=1) max(nu=1,...,n)(h) vertical bar Sigma(n)(k=1) Z(k)(nu)vertical bar <= (h - 1 + o(1)root 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 root logn.