The following question was asked by a colleague of mine. For any prime $p$ consider
$$ M\_p:=\min\_{z\_1,\dots,z\_p}\max\_{j,k}\left|z\_1^k+\dots+z\_j^k\right|,$$
where $z\_1,\dots,z\_p$ are the complex $p$-th roots of unity in any order, and $j,k\in\{1,\dots,p-1\}$ are arbitrary. Is $M\_p$ bounded?



Power sums of p-th roots of unity
This is closely related to the problems surrounding Turán's power sum method. For example see the chapter in Montgomery's Ten Lectures book, or this paper of [Gonek](http://projecteuclid.org/euclid.mmj/1029002459). Lemma 1 (attributed to Cassels) there shows that if $b\_j >0$, and $|z\_j|=1$ (for $j=1$, $\ldots$, $N$) then for $K>N$ 
$$ 
\max\_{k\le K} \Big| \sum\_{j=1}^{N} b\_j z\_j^k \Big| \ge \frac{\sqrt{K-N}}{\sqrt{K}} \Big(\sum\_{j=1}^{N} b\_j^2 \Big)^{\frac 12}. 
$$ 
Now apply this in your situation with $N=(p-1)/2$ (this is $j$ in your problem) and $K=(p-1)$ to get essentially the result you have. Of course your proof is easy enough in this case, but I thought you might appreciate the general context.


