A307216 - OEIS

%I #16 Jan 25 2024 14:31:17

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

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

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

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

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.

%F Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(5*j)/j)).

%F Equals 1/(Gamma(1/(2*phi^2) - i*(5^(1/4)/(2*sqrt(phi)))) * Gamma(phi^2/2 + i*5^(1/4)*(sqrt(phi)/2)) * Gamma(phi^2/2 - i*5^(1/4)*(sqrt(phi)/2)) * Gamma(1/(2*phi^2) + i*(5^(1/4)/(2*sqrt(phi))))), where i is the imaginary unit and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

%e 2.07422504479637891390708968594384056977125337962227288334734036988361960596259...

%p evalf(Product(1 + 1/j^5, j = 1..infinity), 120);

%t RealDigits[Chop[N[Product[(1 + 1/n^5), {n, 1, Infinity}], 120]]][[1]]

%t With[{g = GoldenRatio}, Chop[N[1/(Gamma[1/(2*g^2) - I*5^(1/4)/(2*Sqrt[g])] * Gamma[g^2/2 + I*5^(1/4) * Sqrt[g]/2] * Gamma[g^2/2 - I*5^(1/4) * Sqrt[g]/2] * Gamma[1/(2*g^2) + I*5^(1/4)/(2*Sqrt[g])]), 120]]]

%t N[1/Abs[Gamma[Exp[2*Pi*I/5]]*Gamma[Exp[6*Pi*I/5]]]^2, 120] (* _Vaclav Kotesovec_, Apr 27 2020 *)

%o (PARI) default(realprecision, 120); exp(sumalt(j=1, -(-1)^j*zeta(5*j)/j))

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

%K nonn,cons

%O 1,1

%A _Vaclav Kotesovec_, Mar 29 2019