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,]
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 |
Bir'{o} [Bir00] |
|
| 0.69368 |
Harcos [Bir00] |
|
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Mentioned in [Atk69] in a (presumably unpublished) technical report.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 |
Bir'{o} [Bir94] |
|
| >1/2 |
Bir'{o} [Bir00b] |
Bir'{o}’s proof delivers some computable constant $1/2<c<1/\sqrt{2}$, but it is not computed there exactly which. |
- 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] Bir'{o}, 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] Bir'{o}, A. An upper estimate in Tur'{a}n’s pure power sum problem. Indag. Math. (N.S.) 11 (2000), no. 4, 499-508.
- [Bir00b] Bir'{o}, 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.