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%I #27 Aug 31 2017 03:15:55
%S 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,
%T 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,
%U 0,7,6,8
%N Decimal expansion of the Markoff number asymptotic density constant.
%C 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.
%C 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
%D Richard Guy, "Unsolved Problems in Number Theory" (section D12).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.31.3 Markov-Hurwitz Equation, p. 201.
%D Don B. Zagier, Distribution of Markov numbers, Abstract 796-A37, Notices Amer. Math. Soc. 26 (1979) A-543.
%H Don Zagier, <a href="http://dx.doi.org/10.1090/S0025-5718-1982-0669663-7">On the number of Markoff numbers below a given bound</a>, Mathematics of Computation 39:160 (1982), pp. 709-723.
%H Jean-François Alcover, <a href="/A261613/a261613.txt">Mathematica program</a>
%F C = (3/Pi^2) lim_{N->inf} Sum_{(p,q,r),q<=N<r} 1/(f(p)f(q))
%F = (3/Pi^2) Sum_{(p,q,r)} c(p,q,r)(f(p)+f(q)-f(r))/(f(p)f(q)f(r))
%F 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).
%F The second version demonstrates the rapid convergence on observing that f(p)+f(q)-f(r) = O(1/r^2).
%e C = 0.18071710471180647805779264904916762147630562767088273...
%Y Cf. A002559 (Markoff numbers).
%K nonn,cons,more
%O 0,2
%A _Christopher E. Thompson_, Aug 26 2015
%E Digits to a(72) by using Markoff numbers up to 10^40, from _Christopher E. Thompson_, Aug 28 2015
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