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2c51fa5
Improve C82 upper bound
maaxgrin Jun 1, 2026
e3e4a89
Clarify C82 computation note
maaxgrin Jun 1, 2026
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2 changes: 1 addition & 1 deletion README.md
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Expand Up @@ -117,7 +117,7 @@ Bounds for which the level of available verification is currently at minimal lev
| [79](https://teorth.github.io/optimizationproblems/constants/79a.html) | Asymptotic essential-dimension ratio of the symmetric groups | $\frac{1}{2}$ | 1 |
| [80](https://teorth.github.io/optimizationproblems/constants/80a.html) | Ising perceptron capacity threshold | >0 (0.833*) | 0.847 (0.833*) |
| [81](https://teorth.github.io/optimizationproblems/constants/81a.html) | Brun's constant | 1.840503 | 2.288513 |
| [82](https://teorth.github.io/optimizationproblems/constants/82a.html) | Essential minimum of the Zhang-Zagier height | 0.24874 | 0.25444 |
| [82](https://teorth.github.io/optimizationproblems/constants/82a.html) | Essential minimum of the Zhang-Zagier height | 0.24874 | 0.2536331090204145 |
| [83](https://teorth.github.io/optimizationproblems/constants/83a.html) | Wirsing Constant | 0.30366300 | 0.30366300 |
| [84a](https://teorth.github.io/optimizationproblems/constants/84a.html) | Erdős unit distance exponent | 1.014 (1.03583*) | $\frac{4}{3}\approx 1.3333$ |
| [84b](https://teorth.github.io/optimizationproblems/constants/84b.html) | Sum-product exponent for the reals | $\frac{4}{3}+\frac{10}{4407}\approx 1.3356$ | $<2$ (1.999281*) |
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11 changes: 10 additions & 1 deletion constants/82a.md
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Expand Up @@ -21,7 +21,8 @@ h_Z(\alpha)\le H\}$ is finite.
| ----- | --------- | -------- |
|0.39679 | [D03], Section 7 | |
| 0.25594 | [Doc01a] | $h_Z=\log \mathfrak{h}$ in his notation |
| 0.25444 | [Doc01b] | Best known upper bound |
| 0.25444 | [Doc01b] | Previous best known upper bound |
| 0.2536331090204145 | [Gri26] | Computation based on Doche's polynomials |

## Known lower bounds

Expand Down Expand Up @@ -60,6 +61,10 @@ Math. Comp. 72 (2003), no. 243, 1487–1499
- [F18] Flammang, V.
On the Zhang-Zagier measure.
Int. J. Number Theory 14 (2018), no. 10, 2663–2671
- [Gri26] Max Grinsztajn.
Code and certification data for an improved C82 Zhang-Zagier bound.
GitHub repository, 2026.
https://github.com/maaxgrin/zhang-zagier-c82-bound
- [S80] Smyth, C. J.
On the measure of totally real algebraic integers.
J. Austral. Math. Soc. Ser. A 30 (1980/81), no. 2, 137–149.
Expand All @@ -69,3 +74,7 @@ Math. Comp. 61 (1993), no. 203, 485–491.
- [Zha95] Zhang, S.
Positive line bundles on arithmetic varieties.
J. Amer. Math. Soc. 8 (1995), no. 1, 187–221.

## Contribution notes

Prepared with assistance from GPT-5.5 Pro.