Problem 973

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claims/: The 2 claim pages of Problem 973, one per claimant's result; the problem's standing derives from them.

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Statement. 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

$$\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}?$$

Status. Open. The site's proof-claims tab carries one full proof claim, filed 15 July 2026 by Luo, Yang and Zhu, who name GPT 5.6 Sol Pro on the tab, and the formal-conjectures statement file marks the problem research solved citing their preprint; the site's label is unchanged (OPEN; page last edited 23 January 2026, before the claim). The pending claims, both negative answers, are [[problems/analysis/E0973/claims/2026_07_15_luo_yang_zhu|the page of Luo, Yang and Zhu]] and [[problems/analysis/E0973/claims/2026_08_03_tan_wang_huang_chen|the page of Tan, Wang, Huang and Chen]].

Source. erdosproblems.com/973, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #973, https://www.erdosproblems.com/973.

References.

• [Er92f] Erdős, L., On some problems of P. Turán concerning power sums of complex numbers. Acta Math. Hungar. (1992), 11-24.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
• [Tu84b] Turán, Paul, On a new method of analysis and its applications. (1984), xvi+584.

Formalization. Statement in formal-conjectures (the linked commit, main on 2026-10-07), which marks the problem research solved with the answer no, citing Luo, Yang and Zhu, and leaves its proof unfilled.

Current assessment

The site's formulation asks for a constant $C>1$ such that, for every $n\ge2$, some $z_1,\ldots,z_n$ with $z_1=1$ and $|z_i|\ge1$ have every power sum of index $2$ to $n+1$ below $C^{-n}$ in modulus. The site's commentary records the background: Erdős had such an exponentially small construction when the points are instead confined to the closed unit disk, with $C$ about $1.32$, and L. Erdős [Er92f] located that minimum between $1.746^{-n}$ and $1.745^{-n}$, while Turán's theorem [Tu84b] bounds the maximum below by $(2e)^{-(1+o(1))n}$ when every point has modulus at least one. Turturean's unpublished manuscript of April 2026 (https://github.com/davidturturean/erdos-973), which Luo, Yang and Zhu cite, claims that the normalized infimum of the maximum is at least $\exp(-(0.0597505\ldots+o(1))n)$, so that any admissible constant would satisfy $C\le 1.06157\ldots$; the closing paragraph of its verification appendix records a remaining gap in the proof of its numerical lemma, which the first version of Tan, Wang, Huang and Chen also notes; it has not been reviewed, the site's thread and proof-claims tab do not carry it, and, as a bound that both negative answers below supersede, it is recorded here in prose and has no claim page. Two preprints answer the question negatively. Luo, Yang and Zhu prove that the maximum exceeds $e^{-\lambda n}$ for every fixed $\lambda>0$ once $n$ is large, so the $n$th root of the optimal maximum tends to $1$; Tan, Wang, Huang and Chen prove the sharper bound $\exp(-(1+o(1))\sqrt n\log n)$ (in the third version of their preprint; the first gave a bound with exponent of order $n^{2/3}(\log n)^{2/3}$) through a residual bound for polynomials with zeros in the closed unit disk. Both are recorded as pending full claims on the page of Luo, Yang and Zhu and the page of Tan, Wang, Huang and Chen; neither is refereed, the site has neither changed its label nor credited either paper, and the proofs have not been checked. The frontmatter's standing derives from these two agreeing pending claims.

No Lean proof of the statement has been built or audited here, so neither claim lists formalized evidence. The claimants' repository checks supporting identities and the final contradiction but not the theorem, as its README says and a thread comment notes, and is recorded as code on their page; the formal-conjectures file marks the problem research solved with the answer no on the strength of the preprint, which is a statement file and no formalization; and Linmiao Xu's Lean development, registered in the Palomar registry on 20 September 2026 and following the method of Tan, Wang, Huang and Chen, is linked from their page as a formalization with the registry's own caveats, not built here. The search scope is the site page, its thread and proof-claims tab, the two arXiv records, the formal-conjectures file, the claimants' repository and the Palomar record, as of 2026-10-07; no wider literature search was made.