%I #34 Nov 27 2024 15:47:27

%S 1,2,9,2,3,0,4,1,5,7,1,2,8,6,8,8,6,0,7,1,0,9,1,3,8,3,8,9,8,7,0,4,3,2,

%T 0,6,5,3,4,2,9,6,1,4,2,5,0,1,2,9,9,7,2,4,1,2,2,7,6,2,9,2,3,1,6,1,9,5,

%U 0,0,0,5,5,2,8,2,3,2,0,7,9,4,2,7,3,0,3,0,7,5,9,7,5,5,2,4,4,9,9,4,1,6,1,3,2

%N Decimal expansion of (4/sqrt(Pi))*exp(-gamma/2)*K where K is the Landau-Ramanujan constant and gamma the Euler-Mascheroni constant.

%C An illustration of the Chebyshev effect.

%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 100.

%H Gareth A. Jones and Alexander K. Zvonkin, <a href="https://arxiv.org/abs/2401.00270">A number-theoretic problem concerning pseudo-real Riemann surfaces</a>, arXiv:2401.00270 [math.NT], 2023. See page 5.

%H S. Uchiyama, <a href="http://dx.doi.org/10.1090/S0002-9939-1971-0277494-X">On some products involving primes</a>, Proc. Amer. Math. Soc. 28 (1971) 629-630; MR 43#3227.

%F Equals (4/sqrt(Pi))*exp(-gamma/2)*K = lim_{x->oo} Product_{p prime, p == 1 (mod 4), p <= x} (1 - 1/p).

%F Equals 4*A087197*A064533/exp(A155739). - _R. J. Mathar_, Feb 05 2009

%e 1.2923041571286886071...

%t digits = 105; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 4/Sqrt[Pi]*Exp[-EulerGamma/2]*LandauRamanujanK // RealDigits[#, 10, digits] & // First (* _Jean-François Alcover_, Jun 04 2014, updated Mar 14 2018 *)

%Y Cf. A087197, A064533, A088541, A155739.

%K cons,nonn

%O 1,2

%A _Benoit Cloitre_, Nov 16 2003

%E Offset corrected by _R. J. Mathar_, Feb 05 2009