On some power sum problems of Montgomery and Turan
Abstract
Description
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.
v1: 9 pages; v2: Minor changes. Fixed error in last three lines of proof of Theorem 2: v3: New title. Minor changes
v1: 9 pages; v2: Minor changes. Fixed error in last three lines of proof of Theorem 2: v3: New title. Minor changes