A243378 Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 3 modulo 4} (1 + 1/p).

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, 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, 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

Offset: 0

Author

Jean-François Alcover, Jun 04 2014

Keywords

• nonn
• cons

Examples

			0.985247581006096343690510604298896801...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Eric Weisstein's World of Mathematics, Ramanujan constant.

Crossrefs

Cf. A001620, A064533, A088538, A088540.

Programs

Mathematica

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 *)

Formula

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

Equals 4/(Pi*A088540) = A088538/A088540. - Amiram Eldar, Nov 16 2021