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Turán's pure power sum problem: certified bounds for finite nn and lim sup⁡Rn≤0.68637\limsup R_n\le 0.68637

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CV — Complex Variables

Contributed by Vasileios Bakeas ↗, Ricardo Pinho ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

For n≥1n\ge1 let Rn=inf⁡max⁡1≤k≤n∣z1k+⋯+znk∣R_n=\inf\max_{1\le k\le n}|z_1^k+\dots+z_n^k|, the infimum being over all complex z1=1,z2,…,znz_1=1,z_2,\dots,z_n. This is Turán's pure power sum problem (Erdős problem #519). The best published upper bound for the constant lim sup⁡Rn\limsup R_n is 0.693680.69368, from a computation of Harcos reported by Biró in 2000. We prove lim sup⁡n→∞Rn≤0.68637\limsup_{n\to\infty}R_n\le0.68637. The construction is the Biró–Harcos linearization, with the first half of the power sums prescribed as sk=Teiφ(k/n)s_k=Te^{i\varphi(k/n)} for a piecewise-linear phase φ\varphi that varies from the start. We prove a limit theorem: the quantity that controls the second half converges to an explicit functional of the solution of a linear Volterra equation whose associated operator is a contraction. The resulting inequality is certified in ball arithmetic by an a posteriori error bound, with two separately written programs that follow the same mathematical argument. We also give rigorous upper bounds for RnR_n at 13 values of nn between 300300 and 10610^6, from explicit exact certificates, each checked by one or two programs written independently of the generator. In an appendix, R. Pinho applies the limit theorem to a finer profile, with sharper error bounds, and certifies lim sup⁡Rn≤0.6863607950\limsup R_n\le0.6863607950. The work was done with substantial help from AI models; it has not been refereed.

Provenance statement

Prepared with Claude (Anthropic; Claude Opus 5.5 and Claude Sonnet 5.5) under the author's direction. The AI models were used for the literature search, for designing the constructions, for deriving and writing the proofs, for writing all of the code, and for drafting the text. The author directed the work, ran all computations, and checked the results and the references. A separate AI instance reviewed the proofs and wrote the second checker without access to the first. Sections 1-2 were read by A. Biró and G. Harcos; the rest of the paper has not yet been checked by a professional mathematician. Appendix B was written by Ricardo Pinho with AI agents under his direction; its last paragraph states how it was prepared and checked. The work is unrefereed.

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude SonnetVersion 5.5
FLINT / Arb
python-flintVersion 0.9.0

References

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  2. F.\,V. Atkinson, Some further estimates concerning sums of powers of complex numbers , Acta Math. Acad. Sci. Hungar. 20 (1969), 193--210.
  3. V. Bakeas, Turán's pure power sum constant: C42≤0.68637C_{42}\le0.68637 , version 1.1.0 (2026), https://github.com/Bakeas/turan-c42-bound, doi:10.5281/zenodo.23836302 (https://doi.org/10.5281/zenodo.23836302).DOI
  4. V. Bakeas, Certified upper bounds in Turán's pure power sum problem (Erdős #519): certificates and checkers , version 1.0.0 (2026), https://github.com/Bakeas/turan-power-sum-certificates, doi:10.5281/zenodo.23806909 (https://doi.org/10.5281/zenodo.23806909).DOI
  5. A. Biró, On a problem of Turán concerning sums of powers of complex numbers , Acta Math. Hungar. 65 (1994), 209--216.
  6. A. Biró, An improved estimate in a power sum problem of Turán , Indag. Math. (N.S.) 11 (2000), 343--358.
  7. A. Biró, An upper estimate in Turán's pure power sum problem , Indag. Math. (N.S.) 11 (2000), 499--508.
  8. A.\,Y. Cheer and D.\,A. Goldston, Turán's pure power sum problem , Math. Comp. 65 (1996), 1349--1358.
  9. T. Bloom, Erdős problem #519 , https://www.erdosproblems.com/519.link
  10. S. Griego, An improved asymptotic certificate for Turán's pure power sum constant C42C_{42} , version 1.0.0 (2026), https://github.com/sebastian-griego/turan-c42-certificate.link
Show all 18 referencesShow fewer references
  1. G. Harcos, On a result of András Biró , unpublished manuscript (2000), https://www.renyi.hu/~gharcos/biro.pdf.link
  2. J. Komlós, A. Sárközy and E. Szemerédi, On sums of powers of complex numbers (in Hungarian), Mat. Lapok 15 (1964), 337--347.
  3. F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic , IEEE Trans. Comput. 66 (2017), 1281--1292.
  4. D. Koshta, Lowering the first-block cut-off in the coefficient-cancellation bound for Turán's pure power-sum constant , unrefereed note (2026), https://github.com/DennisKoshta/c42-turan-power-sum-certificate.link
  5. R. Pinho, A 256-step certificate for Turán's pure power-sum constant , unrefereed, Hexagon preprint hexagon:2610.00485v1 (https://hexagonmath.org/2610.00485v1) (2026); code at https://github.com/rpinho/c42-certificate/tree/a8a9bc309bd1992413fa07595bd60c5e3fa1b65b.hexagon
  6. R. Pinho, Certificate for C42≤0.6863607950C_{42}\le0.6863607950 on Bakeas's Theorem A , unrefereed (2026), https://github.com/rpinho/math-certificates/tree/2f49f2f05105d89515c969706600e05b88ae05cf/c42/volterra-quadratic-knots; Optimization Constants repository, pull request #305 (https://github.com/teorth/optimizationproblems/pull/305).link
  7. T. Tao et al., Optimization constants repository , crowdsourced, constant C42, https://github.com/teorth/optimizationproblems.link
  8. P. Turán, On a New Method of Analysis and its Applications , Wiley, New York, 1984.

Version history

  1. v1Submitted by Vasileios BakeasInitial depositCurrentNov 03, 2026