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

0,1

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.

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

Example

0.99984999024682965633806705924046378147600743300742806972498742924...

Mathematica

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

Prog

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

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)).

Sequence in context: A346260 A019898 A330119 * A114054 A347220 A348294

Adjacent sequences: A258812 A258813 A258814 * A258816 A258817 A258818

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jun 11 2015

Status

approved