A243379 Decimal expansion of 1/(2*K^2) = Product_(p prime congruent to 3 modulo 4) (1 - 1/p^2), where K is the Landau-Ramanujan constant.

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

Offset: 0

Author

Jean-François Alcover, Jun 04 2014

Keywords

• nonn
• cons

Comments

Equals 1/1.168075586.., where 1.168.. is zeta_(m=4,n=3)(s=2) in the table of Section 3.3 of arxiv:1008.2547. - R. J. Mathar, Nov 14 2014

Examples

			0.856108981721893476906033006148061173481...
		

References

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

Links

• Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
• Eric Weisstein's MathWorld, Ramanujan constant

Crossrefs

Cf. A002145, A064533, A088539, A243381.

Programs

Mathematica

digits = 101; 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/(2*LandauRamanujanK^2) // RealDigits[#, 10, digits] & // First (* updated Mar 18 2018 *)

Formula

1/(2*K^2), where K is the Landau-Ramanujan constant (A064533).

A088539 * A243379 = 8 / Pi^2. - Vaclav Kotesovec, Apr 30 2020