A175570 Decimal expansion of the Dirichlet beta function of 6.

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

Offset

0,1

References

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308.

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

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 World of Mathematics, Dirichlet Beta Function.

Wikipedia, Dirichlet beta function.

Formula

Equals Sum_{n>=1} A101455(n)/n^6. [see arxiv:1008.2547, L(m=4,r=2,s=6)] [corrected by R. J. Mathar, Feb 01 2018]

Equals (PolyGamma(5, 1/4) - PolyGamma(5, 3/4))/491520. - Jean-François Alcover, Jun 11 2015

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^6)^(-1). - Amiram Eldar, Nov 06 2023

Example

0.998685222218438135441600...

Maple

DirichletBeta := proc(s) 4^(-s)*(Zeta(0, s, 1/4)-Zeta(0, s, 3/4)) ; end proc: x := DirichletBeta(6) ; x := evalf(x) ;

Mathematica

RealDigits[ DirichletBeta[6], 10, 105] // First (* Jean-François Alcover, Feb 11 2013, updated Mar 14 2018 *)

Prog

(PARI) beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
beta(6) \\ Charles R Greathouse IV, Jan 31 2018
(PARI) sumpos(n=1, (12288*n^5 - 30720*n^4 + 33280*n^3 - 19200*n^2 + 5808*n - 728)/(16777216*n^12 - 100663296*n^11 + 270532608*n^10 - 429916160*n^9 + 449249280*n^8 - 324796416*n^7 + 166445056*n^6 - 60899328*n^5 + 15793920*n^4 - 2833920*n^3 + 334368*n^2 - 23328*n + 729), 1) \\ Charles R Greathouse IV, Feb 01 2018

Crossrefs

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

Sequence in context: A346438 A304029 A019896 * A050812 A139345 A231470

Adjacent sequences: A175567 A175568 A175569 * A175571 A175572 A175573

Keyword

cons,easy,nonn

Author

R. J. Mathar, Jul 15 2010

Status

approved