On some problems of P. Turán concerning power sums of complex numbers (Q1207381)

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 \(M_{2,n}\leq(1,745)^{- n}\).