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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

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. 709-723.

Jean-Fran├žois 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^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

Cf. A002559 (Markoff numbers).

Sequence in context: A059679 A198559 A154461 * A165268 A201321 A245737

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

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Last modified September 22 11:02 EDT 2017. Contains 292342 sequences.