

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
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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 MarkovHurwitz Equation, p. 201.
Don B. Zagier, Distribution of Markov numbers, Abstract 796A37, Notices Amer. Math. Soc. 26 (1979) A543.


LINKS

Table of n, a(n) for n=0..71.
Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709723.
JeanFrançois Alcover, Mathematica program


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^24))/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

Cf. A002559 (Markoff numbers).
Sequence in context: A198559 A292887 A154461 * A165268 A306071 A201321
Adjacent sequences: A261610 A261611 A261612 * A261614 A261615 A261616


KEYWORD

nonn,cons,more


AUTHOR

Christopher E. Thompson, Aug 26 2015


EXTENSIONS

Digits to a(72) by using Markoff numbers up to 10^40, from Christopher E. Thompson, Aug 28 2015


STATUS

approved



