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A175572 Decimal expansion of the Dirichlet beta function of 4. 9
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 (list; constant; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..104.

Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Wikipedia, Dirichlet beta function

FORMULA

Equals sum_{n>=1} A101455(n)/n^4. [corrected by R. J. Mathar, Feb 01 2018]

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

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

DirichletBeta[x_] := (Zeta[x, 1/4] - Zeta[x, 3/4])/4^x; RealDigits[ DirichletBeta[4], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

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 (at 1), A006752 (at 2), A153071 (at 3).

Sequence in context: A195477 A157680 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

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Last modified February 23 23:56 EST 2018. Contains 299595 sequences. (Running on oeis4.)