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Level B · Reproducible Analysis P-constant-42a-turan-s-pure-power-sum-constant

Turan's pure power sum constant

The constant C_42 is limsup_n→ inftyR_n, where R_n=minmax_1≤ k≤ n lvert Σ_1≤ i≤ nz_i^krvert, where the minimum is taken over all z_1,…,z_n∈ ℂ with max_i lvert z_irvert=1.

From the catalogue. Imported from Terence Tao and contributors (optimizationproblems repository) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-constant-42a-turan-s-pure-power-sum-constant,
  title        = {Turan's pure power sum constant},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-42a-turan-s-pure-power-sum-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

Description of constant

The constant is , where where the minimum is taken over all with .

Known upper bounds

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

Known lower bounds

BoundReferenceComments
1/6Atkinson [Atk61]
AtkinsonMentioned in [Atk69] in a (presumably unpublished) technical report.
Atkinson [Atk69]
1/2Biró [Bir94]
>1/2Biró [Bir00b]Biró's proof delivers some computable constant , but it is not computed there exactly which.

Additional comments

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

References

  • [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 . GitHub repository, version v1.0.0, commit c8ddce14d9a5e898406d5dc6b8d08bb8a39507c7 (2026). https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0

What counts as progress

  • A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
  • A formal proof (Lean) of a known bound, or a precise error in a claimed one.
  • New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.