A243378 - OEIS

%I #24 Nov 16 2021 07:21:24

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

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

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

%N Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 3 modulo 4} (1 + 1/p).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 101.

%H G. C. Greubel, <a href="/A243378/b243378.txt">Table of n, a(n) for n = 0..5000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Landau-RamanujanConstant.html">Ramanujan constant</a>.

%F Equals (1/sqrt(Pi))*exp(gamma/2)*1/K, where gamma is the Euler-Mascheroni constant (A001620) and K the Landau-Ramanujan constant (A064533).

%F Equals 4/(Pi*A088540) = A088538/A088540. - _Amiram Eldar_, Nov 16 2021

%e 0.985247581006096343690510604298896801...

%t digits = 102; 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]; 1/Sqrt[Pi]*Exp[EulerGamma/2]*1/LandauRamanujanK // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)

%Y Cf. A001620, A064533, A088538, A088540.

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Jun 04 2014