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 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 Don Zagier, On the number of Markoff numbers below a given bound, Mathematics of Computation 39:160 (1982), pp. 709-723. Jean-François Alcover, Mathematica program FORMULA C = (3/Pi^2) lim_{N->inf} Sum_{(p,q,r),q<=N

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