Turan's problem 10 revisited

Details Turan's problem 10 revisited Turan's problem 10 revisited

2006-09-11

arxiv.org/abs/math/0609271v3

Mathematics

Number Theory

v1: 20 pages; v2: 22 pages. Misprints/minor errors fixed. Added some material and some references; v3: 21 pages. Minor errors

Scientific paper

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.

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