A261613 Decimal expansion of the Markoff number asymptotic density constant.

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

Offset: 0

Author

Christopher E. Thompson, Aug 26 2015

Keywords

• nonn
• cons
• more

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

Examples

			C = 0.18071710471180647805779264904916762147630562767088273...
		

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

Crossrefs

Cf. A002559 (Markoff numbers).

Formula

C = (3/Pi^2) lim_{N->inf} Sum_{(p,q,r),q<=N

= (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).

Extensions

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

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