On some problems of P. Turán concerning power sums of complex numbers
type: Item
instance of: scholarly article
title: On some problems of P. Turán concerning power sums of complex numbers
zbMATH Open document ID: 0766.11040
DOI: 10.1007/BF00052086
published in: Acta Mathematica Hungarica
publication date: 1993-04-01T00:00:00Z
review text: Let \(z_ 1,\dots,z_ n\) be complex numbers satisfying \(\max_ j | z_ j|=z_ 1=1\), and let \(S_ k=\sum_{j=1}^ n z_ j^ k\). P. Erdős considered the problem of finding upper and lower bounds for \(M_{m,n}={\displaystyle{\min_ z \max_{m+1\leq k\leq m+n}}} | s_ k|\). It is known that \[ \sqrt{n}(n/(4e(m+n)))^{n-1}\ll M_{m,n}\ll n(n/(4e(m+n))), \] and that \(1/6\leq M_{0,n}\leq 1\). The author has proved in this paper that, for sufficiently large \(n\), \[ n^{-0.7823} \exp(-2\theta n)\leq M_{1,n} \leq n^{4.5} \exp(- 2\theta n) \] where \(\theta\) is the positive root of the equation \(1+\theta+\log \theta=0\). He also proves that \(M_{2,n}\leq(1,745)^{- n}\).
Mathematics Subject Classification ID: 11N30
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 149644
author: László Erdős
reviewed by: G. A. Kolesnik
MaRDI profile type: Publication
cites work: On sums of powers of complex numbers
cites work: Q5342495
cites work: An estimation in the theory of diophantine approximations
cites work: Q3333131
cites work: On some new theorems in the theory of diophantine approximations
full work available at URL: https://doi.org/10.1007/bf00052086
OpenAlex ID: W2148496131
description: scientific article; zbMATH DE number 149644