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Improve C42 upper bound to 0.6906538 #85

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6 changes: 6 additions & 0 deletions constants/42a.md
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Expand Up @@ -13,6 +13,7 @@ where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i
| 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

Expand All @@ -37,3 +38,8 @@ where the minimum is taken over all $z_1,\ldots,z_n\in \mathbb{C}$ with $\max_i
- [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`, commit `c8ddce14d9a5e898406d5dc6b8d08bb8a39507c7` (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.