A258816 Decimal expansion of the Dirichlet beta function of 9.

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

Offset: 0

Author

Jean-François Alcover, Jun 11 2015

Keywords

• nonn
• cons
• easy

Examples

			0.999949684187220089821358873293847527372747996917961601223162723...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
• Wikipedia, Dirichlet beta function.

Crossrefs

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)).

Programs

Magma

SetDefaultRealField(RealField(100)); R:=RealField(); 277*Pi(R)^9/8257536; // G. C. Greubel, Aug 24 2018

Mathematica

RealDigits[DirichletBeta[9], 10, 104] // First

PARI

default(realprecision, 100); 277*Pi^9/8257536 \\ G. C. Greubel, Aug 24 2018

Formula

beta(9) = Sum_{n>=0} (-1)^n/(2n+1)^9 = (zeta(9, 1/4) - zeta(9, 3/4))/262144 = 277*Pi^9/8257536.

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^9)^(-1). - Amiram Eldar, Nov 06 2023