On some power sum problems of Montgomery and Turan

TL;DR

This paper employs character sum estimates over finite fields to resolve open power sum problems posed by Montgomery and Turan, establishing bounds that match the expected magnitude and surpass previous results.

Contribution

It introduces a novel application of Katz's character sum estimates to solve longstanding open problems in power sums, providing tight bounds.

Findings

Established bounds for power sums matching the conjectured order of magnitude.

Improved previous bounds by a factor of sqrt(log n).

Demonstrated the effectiveness of character sum estimates in additive number theory.

Abstract

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.