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A242433 Decimal expansion of one of the Pell-Stevenhagen constants.

%I A242433 #8 Feb 16 2025 08:33:22
%S A242433 2,6,9,7,3,1,8,4,6,1,9,6,9,6,3,3,7,7,3,8,2,1,2,7,1,0,6,7,4,8,9,1,0,8,
%T A242433 1,9,1,9,4,4,7,4,4,4,6,3,5,4,0,4,4,6,4,2,4,8,1,8,1,7,6,7,0,0,1,7,2,5,
%U A242433 8,5,6,9,1,1,3,0,9,7,5,9,0,5,4,9,5,1,2,0,7,2,5,2,0,0,4,7,7,3,9,9
%N A242433 Decimal expansion of one of the Pell-Stevenhagen constants.
%C A242433 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)).
%D A242433 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 119.
%H A242433 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/Landau-RamanujanConstant.html">Landau-Ramanujan Constant</a>
%H A242433 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/PellConstant.html">Pell Constant</a>
%F A242433 (6/Pi^2)*P*K where P is the Pell constant 0.5805775582... and K the Landau-Ramanujan constant 0.7642236535...
%e A242433 0.26973184619696337738212710674891...
%t A242433 (* 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
%Y A242433 Cf. A064533, A141848.
%K A242433 nonn,cons
%O A242433 0,1
%A A242433 _Jean-François Alcover_, May 14 2014