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Decimal expansion of Product_{k>=1} (1 - 1/A002476(k)^3).
6

%I #17 Aug 25 2021 12:58:56

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

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

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

%N Decimal expansion of Product_{k>=1} (1 - 1/A002476(k)^3).

%C In general, for s > 0, Product_{k>=1} (1 + 1/A002476(k)^(2*s+1)) / (1 - 1/A002476(k)^(2*s+1)) = sqrt(3) * (2*Pi)^(2*s + 1) * zeta(2*s + 1) * A002114(s) / ((2^(2*s + 1) + 1) * (3^(2*s + 1) + 1) * (2*s)! * zeta(4*s + 2)).

%C For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) / (1 - 1/A002476(k)^s) = (zeta(s, 1/6) - zeta(s, 5/6))*zeta(s) / ((2^s + 1)*(3^s + 1)*zeta(2*s)).

%C For s > 1, Product_{k>=1} (1 - 1/A002476(k)^s) * (1 - 1/A007528(k)^s) = 6^s / ((2^s - 1)*(3^s - 1)*zeta(s)).

%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 3 1 3 = 1/A334478).

%F A334477 / A334478 = 15*sqrt(3)*zeta(3)/Pi^3.

%F A334478 * A334480 = 108/(91*zeta(3)).

%e 0.996401692816036632262361122384718799965573818714...

%Y Cf. A002476, A175646, A334425, A334427, A334477.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, May 02 2020

%E More digits from _Vaclav Kotesovec_, Jun 27 2020