Problem 973

Erdős Problem #973: declared status 'open'. Formalized: yes. Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq 1$ for all $1\leq i\leq n$ with$$\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}?$$ Current best: Erd\H{o}s proved (as described on p.35 of [Tu84b]) that such a sequence does exist with $\lvert z_i\rvert\leq 1$. In [Er92f] (a different) Erd\H{o}s refines this analysis, proving that if$$M_2=\min_{z_j} \max_{2\leq k\leq n+1} \left\lvert \sum_{1\leq j\leq n}z_j^k\right\rvert,$$where the minimum is take over all $z_j\in \mathbb{C}$ with $\max \lvert z_j\rvert=1$, then$$(1.746)^{-n} < M_2 < (1.745)^{-n}.$$Tang notes in the comments that Theorem 6.1 of [Tu84b] implies that, if $\lvert z_i\rvert \geq 1$ for all $i$, then$$\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert \geq (2e)^{-(1+o(1))n}.$$See also [519]. Prize: no. Tags: analysis.