Turan's problem 10 revisited
arXiv:math/0609271
Abstract
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 of being a prime we have previously proved the much sharper result that the quantity lies in the interval [sqrt(n),sqrt(n+1)] The method of proof combines a general lower bound (of Andersson), explicit arithmetical constructions (of Montgomery, Fabrykowski or Andersson), moments (probabilistic methods) and estimates for the difference of consecutive primes (of Baker, Harman and Pintz). We also prove some (conditional and unconditional) related results.
v1: 20 pages; v2: 22 pages. Misprints/minor errors fixed. Added some material and some references; v3: 21 pages. Minor errors fixed. Changed problem section and added reference to new paper where we solve some of the open problems of v2