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. 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.