Turan's problem 10 revisited

TL;DR

This paper improves bounds on a problem related to sums of complex roots of unity, combining various mathematical techniques to achieve tighter estimates and exploring related results.

Contribution

It provides a new upper bound for a classical problem, improving previous results and employing a novel combination of methods including arithmetical constructions and probabilistic estimates.

Findings

Improved bound: inf_{|z_k|=1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n + O(n^{0.2625+epsilon})

Special case for prime n+1: the sum lies in [sqrt(n), sqrt(n+1)]

Use of diverse methods: lower bounds, explicit constructions, moments, and prime gap estimates

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.