User:Christopher E. Thompson/sandbox

Testing reference
This reference.

Value for C in asymptotic formula
The value C = 2.3523418721 quoted in the asymptotic formula for $$m_n$$ was computed from the value 0.18071704711507 for C^{-2} in Zagier's 1982 paper. Unfortunately that value is inaccurate, most importantly due to an accidentally missing digit, and in fact a more accurate value is used to compute other numbers in that paper. The value should be 0.180717104711806 for C^{-2} and hence C = 2.3523414972 (to 10 places). See for more details.

I propose to update the main page accordingly in the absence of any objections! Chris Thompson (talk) 20:05, 20 April 2016 (UTC)

Status of the asymptotic formula
The main page states that the conjecture that $$\textstyle m_n = \tfrac13 e^{C\sqrt{n}+o(1)}$$ was "proved by Greg McShane and Igor Rivin in 1995". I think this is incorrect. That conjecture is equivalent to
 * $$M(x) = C(\log(3x))^2 + o(\log x)$$

(not the same C) where $$M(x)$$ is the number of Markoff numbers less than $$x$$. Don Zagier's 1982 paper proved this with an error term of $$\textstyle O(\log x (\log\log x)^2)$$. The McShane and Rivin paper referenced (which incidentally is most easily accessed as ) improves this to $$\textstyle O(\log x \log\log x)$$, which is still way short of $$\textstyle o(\log x)$$. In fact such improved estimates of the error term are clearly described as conjectures, even prompting the authors to remark at the end of the French extended abstract "Nous croyons que la deuxieme conjecture est tres difficile."

In fact the conjecture appears to be still open. The recent paper by Gamburd, Magee and Ronan states (Theorem 1) that "The best current result is due to McShane and Rivin" and gives a reference to their companion 1995 paper, accessible as.

More tests
Using  1 048 576 .0001

Trying as user page reference.

Chemical element such as Tellurium.

The time is now but it was 15:12 when I wrote this.

Trying a signature.Chris Thompson (talk) 15:29, 23 October 2019 (UTC)

References for updates to Julia set
I am not sure how to deal with all the misapprehensions here, but to take the last point, the usage "Fatou set" to denote the complement of the Julia set has become standard in recent decades. Admittedly, it isn't quite as ancient. Writing in 1990, Beardon says
 * Although the use of the term "Julia set" is standard, the use of "Fatou set" was suggested as late as 1984 (in ). It seems appropriate, but the reader should be aware of the common alternatives. namely the stable set, and the set of normality.