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Turan's pure power sum constant (C42C_{42})

Turan's pure power sum constant: determine C42C_{42} or improve its bounds; the repository records >0.5 (lower) and 0.69368 (0.6906538*) (upper).

MSC 11N30 · Turán theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

The constant C42C_{42} is lim sup⁡n→∞Rn\limsup_{n\to \infty}R_n, where

Rn=min⁡max⁡1≤k≤n∣∑1≤i≤nzik∣,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 z1,…,zn∈Cz_1,\ldots,z_n\in \mathbb{C} with max⁡i∣zi∣=1\max_i \lvert z_i\rvert=1.

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Background

  • Computational investigations by Cheer and Goldston [CG96] suggest that C42C_{42} is close to 0.70.7.
  • C42C_{42} is the optimal constant for Erdős problem #519.

References (from the repository page)

  • [Atk61] Atkinson, F. V. On sums of powers of complex numbers. Acta Math. Acad. Sci. Hungar. 12 (1961), 185-188.
  • [Atk69] Atkinson, F. V. Some further estimates concerning sums of powers of complex numbers. Acta Math. Acad. Sci. Hungar. 20 (1969), 193-210.
  • [Bir94] Biró, A. On a problem of Tur'{a}n concerning sums of powers of complex numbers. Acta Math. Hungar. 65 (2000), no. 3, 209-216.
  • [Bir00] Biró, A. An upper estimate in Tur'{a}n's pure power sum problem. Indag. Math. (N.S.) 11 (2000), no. 4, 499-508.
  • [Bir00b] Biró, A. An improved estimate in a power sum problem of Tur'{a}n. Indag. Math. (N.S.) 11 (2000), no. 3, 343-358.
  • [CG96] A. Y. Cheer and D. A. Goldston Tur'{a}n's pure power sum problem. Math. Comp. 65 (1996), no. 215, 1349-1358.
  • [Gri26] Griego, S. An improved asymptotic certificate for Turan's pure power sum constant C42C_{42}. GitHub repository, version v1.0.0, commit c8ddce14d9a5e898406d5dc6b8d08bb8a39507c7 (2026). https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound >0.5; best known upper bound 0.69368 (0.6906538*). Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
1 Trivial
5/6 Biró [Bir00]
0.69368 Harcos [Bir00]
0.6906538* Griego [Gri26] Proposed asymptotic two-block certificate with exact rational interval verification of the limiting inequality. No explicit finite threshold NN is supplied.

Known lower bounds

Bound Reference Comments
1/6 Atkinson [Atk61]
1/31/3 Atkinson Mentioned in [Atk69] in a (presumably unpublished) technical report.
π/8\pi/8 Atkinson [Atk69]
1/2 Biró [Bir94]
>1/2 Biró [Bir00b] Biró's proof delivers some computable constant 1/2<c<1/21/2<c<1/\sqrt{2}, but it is not computed there exactly which.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Turan's pure power sum constant ($C_{42}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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