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on Turan's pure power sum constant ($C_{42}$)

The statement

The constant $C_{42}$ is $\limsup_{n\to \infty}R_n$ , where

R_n=\min\max_{1\leq k\leq n} \left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert,

where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i \lvert z_i\rvert=1$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.