login
A175572
Decimal expansion of the Dirichlet beta function of 4.
14
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
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.
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
KEYWORD
cons,easy,nonn
AUTHOR
R. J. Mathar, Jul 15 2010
STATUS
approved