Turan’s pure power sum constant
Description of constant
The constant $C_{42}$ is $\limsup_{n\to \infty}R_n$, where \(R_n=\min\max_{1\leq k\leq n} \left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert,\) where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i \lvert z_i\rvert=1$.
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 $N$ is supplied. |
| 0.6888863712* | Pinho [Pin26] | Exact $215276991/312500000$; 256 uniform middle blocks with $t_j=(256+j)/768$ and $\alpha=0.64427390171288+0.58993506009068i$. Rational profile: SHA-256 705a729155eb508878d17ebea0322627a0ae02451c044a346a99d2e81c7afcab. Separate interval (100 decimal places, 240 terms) and Arb (768 bits, 400 terms) checks give $\lvert Y\rvert/D<0.688886370983466$. Unrefereed analytic asymptotic reduction; no explicit finite threshold, optimality or formal verification claimed. |
| 0.6873358063* | Koshta [Kos2026] | Exact $6873358063/10000000000$. The coefficient-cancellation construction of [Gri26] (with the step profiles of Röhrig, Plata and PR #251), but with the first-block cut-off lowered from $\tau=1/3$ to $\tau=41/200$. Then up to four corrections reach $b_n$, and the limiting quantity gains the terms $-\frac16\sum_{i,j,k}\delta_i\delta_j\delta_k T_3+\frac1{24}\sum_{i,j,k,l}\delta_i\delta_j\delta_k\delta_l T_4$, where $T_d(u)=\int_{x_i\ge u_i,\ \sum x_i\le 1}(1-\sum x_i)^{\alpha-1}/\prod x_i\,dx$; the criterion of PR #251 holds verbatim for every $\tau\in(0,1/2)$ with the corresponding $d$-fold terms ($d<1/\tau$). 64 uniform middle blocks on $[41/200,159/200]$, $\alpha=0.6928404180017845+0.6148849499318954i$. Rational profile SHA-256 be0c406fb2bf58f18a3ecb070daab97b662ec63318d9a83606c2117b8e16cbef. $T_4$ and most $T_3$ are evaluated by exact-rational Dirichlet series with explicit tail bounds, the remaining $T_3$ by a validated 1-D integral; Arb (384 and 640 bits) gives $\lvert Y\rvert/D<0.68733580619$, and an independently written checker agrees. Unrefereed analytic asymptotic reduction; no explicit finite threshold, optimality or formal verification claimed. |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| 1/6 | Atkinson [Atk61] | |
| $1/3$ | Atkinson | Mentioned in [Atk69] in a (presumably unpublished) technical report. |
| $\pi/8$ | Atkinson [Atk69] | |
| 1/2 | Biró [Bir94] | |
| >1/2 | Biró [Bir00b] | Biró’s proof delivers some computable constant $1/2<c<1/\sqrt{2}$, but it is not computed there exactly which. |
Additional comments
- Computational investigations by Cheer and Goldston [CG96] suggest that $C_{42}$ is close to $0.7$.
- $C_{42}$ 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 $C_{42}$. GitHub repository, version
v1.0.0, commitc8ddce14d9a5e898406d5dc6b8d08bb8a39507c7(2026). https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0 -
[Kos2026] Koshta, Dennis, with Claude (Anthropic, Claude Opus 5.5). Lowering the first-block cut-off in the coefficient-cancellation bound for Turán’s pure power-sum constant. Unrefereed note, exact certificate and checkers: https://github.com/DennisKoshta/c42-turan-power-sum-certificate (commit
8e1797e0d35454ad18cfebbbeb5858bf202b0748, 23 Oct 2026). - [Pin26] Pinho, R. A 256-step certificate for Turán’s pure power-sum constant. Unrefereed note (22 October 2026), paper, exact profile and two separately implemented checkers: https://github.com/rpinho/c42-certificate/tree/a8a9bc309bd1992413fa07595bd60c5e3fa1b65b
Contribution notes
The [Gri26] certificate repository and this update were prepared with AI assistance for formatting, exposition, and verification scripts. The mathematical claim, constants, references, and computations are provided for independent review.
The [Pin26] construction refines the step profiles of Griego, André Röhrig and Kerrigan Plata while retaining their coefficient-cancellation method. AI use and author review: Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared the certificate, checkers and submission text. This does not claim a line-by-line human verification. The [Kos2026] row was found and prepared with Claude (Anthropic, Claude Opus 5.5), directed by Dennis Koshta, who takes responsibility for the submission: the general-cut-off extension of the [Gri26] criterion and its proof, the parameter search, the certificate, the Arb checker, and the text. The proof was reviewed by two further Claude agents working from the note and the PR #251 paper only, and the certificate was checked by another Claude agent with its own checker, written from the certificate format specification without access to the search code. No human mathematician has checked the argument.