Power sums of $n$ complex numbers outside the unit disc: can all of them be exponentially small? (Erdős #973)
Statement
Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_1,\dots,z_n\in\mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert\geq 1$ for all $1\leq i\leq n$, such that $$\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n} z_i^k \right\rvert < C^{-n}\,?$$
Acceptance. FULLY RESOLVES: (YES) an explicit construction — a family of configurations $z_1^{(n)},\dots,z_n^{(n)}$ given by formulas or an explicit algorithm, with $z_1=1$ and all $\lvert z_i\rvert\ge 1$ — together with a proof valid for all $n\ge 2$ that $\max_{2\le k\le n+1}\lvert\sum_i z_i^k\rvert<C^{-n}$ for a stated constant $C>1$; or (NO) a proof that for every $C>1$ there are infinitely many $n$ with no such configuration. Machine-checkable proof preferred, else full written proof. ADVANCES: rigorous certified computations (exact or interval arithmetic) of the minimax value $\min\max_{2\le k\le n+1}\lvert s_k\rvert$ under the stated constraints for a growing range of $n$, establishing its empirical exponential rate with reproducible code; a proof of the YES statement for a restricted configuration class; or an improvement of the $(2e)^{-(1+o(1))n}$ lower-bound barrier stated in the background. Deliver the construction + proof, or the optimization/certification code with the attained values and configurations.
Background
A problem of Erdős [Er65b, p.213], recorded as Problem 7.3 in Hayman's 'Research problems in function theory' [Ha74] and as Problem 2.39 in [Va99]; listed as open on erdosproblems.com/973 (fetched 2026-07-13, status 'open', tagged 'analysis'). It belongs to Turán's power-sum theory, where the power sums $s_k=\sum_i z_i^k$ over the window $k=2,\dots,n+1$ measure how much cancellation $n$ unimodular-or-larger numbers can sustain. What is known: with the RELAXED constraint $\lvert z_i\rvert\le 1$ (normalized so $\max\lvert z_j\rvert=1$), Erdős proved such exponentially small configurations exist — his construction gives $C\approx 1.32$ (described on p.35 of Turán's book [Tu84b]). A different Erdős [Er92f] sharpened that regime: with $M_2=\min_{z_j}\max_{2\le k\le n+1}\lvert\sum_j z_j^k\rvert$ over configurations with $\max\lvert z_j\rvert=1$, one has $(1.746)^{-n}<M_2<(1.745)^{-n}$. For the actual question (all $\lvert z_i\rvert\ge 1$), Tang notes on the site that Theorem 6.1 of [Tu84b] gives the barrier $\max_{2\le k\le n+1}\lvert s_k\rvert\ge (2e)^{-(1+o(1))n}$, so any admissible $C$ satisfies $C\le 2e$; whether ANY $C>1$ is attainable when the points are pushed outside the unit circle is the open question. See also Erdős #519 (erdosproblems.com/519). Closely related to (but distinct from) the venue problem on the best constant in the Turán–Atkinson power-sum inequality, which is Problem 7.4 of the same Hayman collection. The attacker's tool: numerical minimax optimization over configurations $\{z_i\}$ with $\lvert z_i\rvert\ge 1$, $z_1=1$ for growing $n$ — estimate the decay rate $(\max_k\lvert s_k\rvert)^{1/n}$, detect a structured extremal family, then certify it rigorously; interval-arithmetic certificates for specific $n$ calibrate any conjectured $C$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #973 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #519 — related power-sum problem | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.