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Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^5).
7

%I #22 Jun 27 2020 11:51:41

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

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

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

%N Decimal expansion of Product_{k>=1} (1 - 1/A002144(k)^5).

%C In general, for s>0, Product_{k>=1} (1 + 1/A002144(k)^(2*s+1))/(1 - 1/A002144(k)^(2*s+1)) = Pi^(2*s+1) * A000364(s) * zeta(2*s+1) / ((2^(2*s+2) + 2) * (2*s)! * zeta(4*s+2)). - _Dimitris Valianatos_, May 01 2020

%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)).

%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 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 5 = 1/A334450).

%F A334449 / A334450 = 4725*zeta(5)/(16*Pi^5).

%F A334450 * A334452 = 32/(31*zeta(5)).

%e 0.999676527079626662018246180873083701500751574379554430568432840424975981921219...

%Y Cf. A002144, A088539, A334446, A334450.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Apr 30 2020

%E More digits from _Vaclav Kotesovec_, Jun 27 2020