A175571 Decimal expansion of the Dirichlet beta function of 5.

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

Offset

0,1

Comments

The value of the Dirichlet L-series L(m=4,r=2,s=4), see arXiv:1008.2547.

References

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

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See p. 240.

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 5*Pi^5/1536 = Sum_{n>=1} A101455(n)/n^5, where Pi^5 = A092731. [corrected by R. J. Mathar, Feb 01 2018]

Equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - Jean-François Alcover, Mar 29 2013

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^5)^(-1). - Amiram Eldar, Nov 06 2023

Example

0.99615782807708806400631936...

Maple

DirichletBeta := proc(s) 4^(-s)*(Zeta(0, s, 1/4)-Zeta(0, s, 3/4)) ; end proc: x := DirichletBeta(5) ; x := evalf(x) ;

Mathematica

RealDigits[ DirichletBeta[5], 10, 105] // First (* Jean-François Alcover, Feb 20 2013, updated Mar 14 2018 *)

Prog

(PARI) 5*Pi^5/1536 \\ Charles R Greathouse IV, Jan 31 2018
(PARI) beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
beta(5) \\ Charles R Greathouse IV, Jan 31 2018

Crossrefs

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

Sequence in context: A021505 A346930 A386518 * A388876 A019894 A389778

Adjacent sequences: A175568 A175569 A175570 * A175572 A175573 A175574

Keyword

cons,easy,nonn

Author

R. J. Mathar, Jul 15 2010

Status

approved