A175572 Decimal expansion of the Dirichlet beta function of 4.

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

Offset

0,1

Comments

This is the value of the Dirichlet L-series for A101455 at s=4, see arXiv:1008.2547, L(m=4,r=2,s=4).

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

Mark W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064, proposition 5.

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^4. [corrected by R. J. Mathar, Feb 01 2018]

Equals (PolyGamma(3, 1/4) - PolyGamma(3, 3/4))/1536. - Jean-François Alcover, Jun 11 2015

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^4)^(-1). - Amiram Eldar, Nov 06 2023

Example

0.988944551741105336108422633...

Maple

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

Mathematica

RealDigits[ DirichletBeta[4], 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(4) \\ Charles R Greathouse IV, Jan 31 2018

Crossrefs

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

Sequence in context: A305382 A347199 A011228 * A263984 A021095 A090998

Adjacent sequences: A175569 A175570 A175571 * A175573 A175574 A175575

Keyword

cons,easy,nonn

Author

R. J. Mathar, Jul 15 2010

Status

approved