On some power sum problems of Montgomery and Turan

@article{Andersson2007OnSP,
  title={On some power sum problems of Montgomery and Turan},
  author={John Andersson},
  journal={arXiv: Number Theory},
  year={2007}
}
  • J. Andersson
  • Published 28 June 2007
  • Mathematics, Computer Science
  • arXiv: Number Theory
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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In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case
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