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Decimal expansion of Product_{n>=1} (1+1/n^6).
9

%I #15 Aug 30 2024 15:48:41

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

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

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

%N Decimal expansion of Product_{n>=1} (1+1/n^6).

%C From _Vaclav Kotesovec_, Aug 30 2024: (Start)

%C For m>0, Product_{k>=1} (1 + m/k^6) = (cosh(Pi*m^(1/6)) - cos(sqrt(3)*Pi*m^(1/6))) * sinh(Pi*m^(1/6)) / (2*Pi^3*sqrt(m)).

%C If m tends to infinity, Product_{k>=1} (1 + m/k^6) ~ exp(2*Pi*m^(1/6)) / (8*Pi^3*sqrt(m)). (End)

%F Equals (cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3).

%F Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(6*j)/j)). - _Vaclav Kotesovec_, Mar 28 2019

%e 2.03474083500942906358682080964285089771090100623925469055753948...

%p evalf((cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3), 120);

%t RealDigits[(Cosh[Pi]-Cos[Sqrt[3]*Pi])*Sinh[Pi]/(2*Pi^3),10,120][[1]]

%Y Cf. A109219, A175615, A175617, A175619, A156648, A073017, A258870, A334411.

%K nonn,cons

%O 1,1

%A _Vaclav Kotesovec_, Jun 13 2015