|
| |
|
|
A261613
|
|
Decimal expansion of the Markoff number asymptotic density constant.
|
|
3
|
|
|
|
1, 8, 0, 7, 1, 7, 1, 0, 4, 7, 1, 1, 8, 0, 6, 4, 7, 8, 0, 5, 7, 7, 9, 2, 6, 4, 9, 0, 4, 9, 1, 6, 7, 6, 2, 1, 4, 7, 6, 3, 0, 5, 6, 2, 7, 6, 7, 0, 8, 8, 2, 7, 3, 4, 8, 0, 5, 3, 8, 8, 8, 9, 6, 6, 5, 0, 5, 6, 0, 7, 6, 8
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
If M(x) is the number of Markoff numbers (A002559) less than x, then Zagier proved that M(x) = C(log(3x))^2 + O(log x (log log x)^2), where the constant C is the value of a rapidly converging sum defined in term of the Markoff numbers themselves. Numerical results suggest that the true error term is substantially smaller.
The value of C (0.18071704711507) published in Zagier's 1982 paper suffers from a missing digit and some rounding errors. However his earlier 1979 abstract has a value (0.180717105) that is correct to 9 decimal places. - Christopher E. Thompson, Oct 05 2015
|
|
|
REFERENCES
|
Richard Guy, "Unsolved Problems in Number Theory" (section D12).
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.31.3 Markov-Hurwitz Equation, p. 201.
Don B. Zagier, Distribution of Markov numbers, Abstract 796-A37, Notices Amer. Math. Soc. 26 (1979) A-543.
|
|
|
LINKS
|
|
|
|
FORMULA
|
C = (3/Pi^2) lim_{N->inf} Sum_{(p,q,r),q<=N<r} 1/(f(p)f(q))
= (3/Pi^2) Sum_{(p,q,r)} c(p,q,r)(f(p)+f(q)-f(r))/(f(p)f(q)f(r))
where the sums are over Markoff triples (p,q,r) with p<=q<=r, c(p,q,r)=1 except for c(1,1,1)=c(1,1,2)=1/2 and f(x) = log ((3x+sqrt(9x^2-4))/2) = arc cosh (3x/2).
The second version demonstrates the rapid convergence on observing that f(p)+f(q)-f(r) = O(1/r^2).
|
|
|
EXAMPLE
|
C = 0.18071710471180647805779264904916762147630562767088273...
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
