A175570 - OEIS

%I #35 Nov 07 2023 04:56:26

%S 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,

%T 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,

%U 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

%N Decimal expansion of the Dirichlet beta function of 6.

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

%H G. C. Greubel, <a href="/A175570/b175570.txt">Table of n, a(n) for n = 0..10000</a>

%H Richard J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_beta_function">Dirichlet beta function</a>.

%F 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]

%F Equals (PolyGamma(5, 1/4) - PolyGamma(5, 3/4))/491520. - _Jean-François Alcover_, Jun 11 2015

%F Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^6)^(-1). - _Amiram Eldar_, Nov 06 2023

%e 0.998685222218438135441600...

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

%t RealDigits[ DirichletBeta[6], 10, 105] // First (* _Jean-François Alcover_, Feb 11 2013, updated Mar 14 2018 *)

%o (PARI) beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x

%o beta(6) \\ _Charles R Greathouse IV_, Jan 31 2018

%o (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

%Y 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)).

%Y Cf. A101455.

%K cons,easy,nonn

%O 0,1

%A _R. J. Mathar_, Jul 15 2010