OFFSET
0,1
COMMENTS
P. Stevenhagen conjectured that the asymptotic counting function of the squarefree integers for which the negative Pell equation x^2 - n*y^2 = -1 has an integer solution, was f(n) ~ (6/Pi^2)*P*K*n/sqrt(log(n)).
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 119.
LINKS
Eric Weisstein's MathWorld, Landau-Ramanujan Constant
Eric Weisstein's MathWorld, Pell Constant
FORMULA
(6/Pi^2)*P*K where P is the Pell constant 0.5805775582... and K the Landau-Ramanujan constant 0.7642236535...
EXAMPLE
0.26973184619696337738212710674891...
MATHEMATICA
(* After Victor Adamchik *) LandauRamanujan[n_] := With[{K = Ceiling[Log[2, n*Log[3, 10]]]}, N[Product[(((1 - 2^(-2^k))*4^2^k*Zeta[2^k])/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^2^(-k - 1), {k, 1, K}]/Sqrt[2], n]]; K = LandauRamanujan[100]; P = 1 - QPochhammer[1/2, 1/4]; RealDigits[6/Pi^2*P*K, 10, 100] // First
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, May 14 2014
STATUS
approved