On some power sum problems of Montgomery and Turan

Details On some power sum problems of Montgomery and Turan On some power sum problems of Montgomery and Turan

2007-06-28

arxiv.org/abs/0706.4131v3

Mathematics

Number Theory

v1: 9 pages; v2: Minor changes. Fixed error in last three lines of proof of Theorem 2: v3: New title. Minor changes

Scientific paper

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.

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