Added 41a.md - Turan's pure power sum constant

constants/41a.md

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+# Turan's pure power sum constant
+
+## Description of constant
+
+The constant $C_{41}$ 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+| $1/3$ | Atkinson | Mentioned in [Atk69] in a (presumably unpublished) technical report.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. |
+
+## Additional comments
+
+Computational investigations by Cheer and Goldston [CG96] suggest that $C_{41}$ is close to $0.7$. This problem is discussed at www.erdosproblems.com/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.