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Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).
8

%I #22 Jun 27 2020 11:52:58

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

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

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

%N Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^4).

%C In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

%C For s>1, zeta(s, 1/4) - zeta(s, 3/4) = (-1)^s*(PolyGamma(s-1, 1/4) - PolyGamma(s-1, 3/4))/(s-1)! = 2*(-1)^s * PolyGamma(s-1, 1/4) / Gamma(s) - 2^s*(2^s - 1)*zeta(s) = 4^s * DirichletBeta(s).

%D B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

%H Ph. Flajolet and I. Vardi, <a href="http://algo.inria.fr/flajolet/Publications/landau.ps">Zeta function expansions of some classical constants</a>, Feb 18 1996, p. 7-8.

%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>

%H R. J. Mathar, <a href="http://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, p. 26 (case 4 1 4 = 1/A334446).

%F A334445 / A334446 = 35*(PolyGamma(3, 1/4)/(8*Pi^4) - 1)/34.

%F A334446 * A334448 = 96/Pi^4.

%e 0.998350495723200406499905517565541629191539407019605795046314...

%Y Cf. A002144, A088539, A334425, A334450.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Apr 30 2020

%E More digits from _Vaclav Kotesovec_, Jun 27 2020