A334411 - OEIS

%I #14 Aug 31 2024 04:42:03

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

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

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

%N Decimal expansion of Product_{k>=1} (1 + 1/k^8).

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

%C For m>0, Product_{k>=1} (1 + m/k^8) = (cosh(Pi*sqrt(2 - sqrt(2))*m^(1/8)) - cos(Pi*sqrt(2 + sqrt(2))*m^(1/8))) * (cosh(Pi*sqrt(2 + sqrt(2))*m^(1/8)) - cos(Pi*sqrt(2 - sqrt(2))*m^(1/8)))/(4*sqrt(m)*Pi^4).

%C If m tends to infinity, Product_{k>=1} (1 + m/k^8) ~ exp(Pi*sqrt(2*(2 + sqrt(2)))*m^(1/8)) / (16*Pi^4*sqrt(m)).

%C In general, if m tends to infinity and v > 2, Product_{k>=1} (1 + m/k^v) ~ exp(Pi*m^(1/v)/sin(Pi/v)) / ((2*Pi)^(v/2)*sqrt(m)). (End)

%F Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(8*j)/j)).

%F Equals (cos(sqrt(4 - 2*sqrt(2))*Pi) + cos(sqrt(4 + 2*sqrt(2))*Pi) + cosh(sqrt(4 - 2*sqrt(2))*Pi) + cosh(sqrt(4 + 2*sqrt(2))*Pi) - 2*cos(sqrt(2 - sqrt(2))*Pi) * cosh(sqrt(2 - sqrt(2))*Pi) - 2*cos(sqrt(2 + sqrt(2))*Pi) * cosh(sqrt(2 + sqrt(2))*Pi)) / (8*Pi^4).

%e 2.00815605499274531514903948232341369211953215983095097877074299617422...

%p evalf(Product(1 + 1/j^8, j = 1..infinity), 120);

%t RealDigits[Chop[N[Product[(1 + 1/n^8), {n, 1, Infinity}], 120]]][[1]]

%o (PARI) default(realprecision, 120); exp(sumalt(j=1, -(-1)^j*zeta(8*j)/j))

%Y Cf. A156648, A073017, A258870, A307216, A258871.

%Y Cf. A109219, A175615, A175616, A175617, A175619.

%K nonn,cons

%O 1,1

%A _Vaclav Kotesovec_, Apr 27 2020