Turan's book *On a new method of analysis and its applications* focuses on bounds on power sums. The quantity
$$
T(m,n)=\inf_{|z_k|=1} \max_{\nu=1,\ldots,m} \left| \sum_{k=1}^n z_k^\nu\right|,
$$
for various choices of $m,n$ has been of interest since then. The case $m\sim n^{B}$ has recently been settled by Andersson, using a character sum estimate due to Katz, in the paper available [on arXiv here][1]. The result essentially states that
$$
T(m,n)\asymp \sqrt{n},
$$
if $m=\lfloor n^B \rfloor,$ if $B>1$ is fixed. This was also an open problem by Montgomery in his *Ten lectures on the interface between analytic number theory and harmonic analysis.*


  [1]: https://arxiv.org/pdf/0706.4131.pdf