Turan's problem 10 revisited

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 $n+1$ 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.