A258815 Decimal expansion of the Dirichlet beta function of 8.

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

Offset: 0

Author

Jean-François Alcover, Jun 11 2015

Keywords

• nonn
• cons
• easy

Examples

			0.99984999024682965633806705924046378147600743300742806972498742924...
		

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)), A258816 (beta(9)).

Programs

Mathematica

RealDigits[DirichletBeta[8], 10, 102] // First

PARI

(zetahurwitz(8,1/4)-zetahurwitz(8,3/4))*(1/4)^8 \\ Hugo Pfoertner, Feb 07 2020

Formula

beta(8) = Sum_{n>=0} (-1)^n/(2n+1)^8 = (zeta(8, 1/4) - zeta(8, 3/4))/65536 = (PolyGamma(7, 1/4) - PolyGamma(7, 3/4))/330301440.

Equals ClausenFunction(8, Pi/2).

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^8)^(-1). - Amiram Eldar, Nov 06 2023