From 2a1f973df0e6902d36cfada389e6beaec283fa28 Mon Sep 17 00:00:00 2001 From: TFBloom <96446181+TFBloom@users.noreply.github.com> Date: Sun, 1 Feb 2026 08:24:18 +0000 Subject: [PATCH] Create 41a.md --- constants/41a.md | 38 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 38 insertions(+) create mode 100644 constants/41a.md diff --git a/constants/41a.md b/constants/41a.md new file mode 100644 index 0000000..468e361 --- /dev/null +++ b/constants/41a.md @@ -0,0 +1,38 @@ +# 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| --------- | -------- | +| 1/6 | Atkinson [Atk61] | | +| $1/3$ | Atkinson | Mentioned in [Atk69] in a (presumably unpublished) technical report.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