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. |
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| $1/3$ | Atkinson | Mentioned in [Atk69] in a (presumably unpublished) technical report.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. Aca 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
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.