TITLE: Power sums of p-th roots of unity
QUESTION [7 upvotes]: 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?

REPLY [7 votes]: 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. 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.