SOLVED
Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that
\[\max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c?\]
A problem of Turán, who proved that this maximum is $\gg 1/n$. This was solved by Atkinson
[At61b], who showed that $c=1/6$ suffices. This has been improved by Biró, first to $c=1/2$
[Bi94], and later to an absolute constant $c>1/2$
[Bi00]. Based on computational evidence it is likely that the optimal value of $c$ is $\approx 0.7$.
