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A307216
Decimal expansion of Product_{k>=1} (1 + 1/k^5).
2
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, 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, 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
OFFSET
1,1
LINKS
Eric Weisstein's World of Mathematics, Infinite Product.
FORMULA
Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(5*j)/j)).
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.
EXAMPLE
2.07422504479637891390708968594384056977125337962227288334734036988361960596259...
MAPLE
evalf(Product(1 + 1/j^5, j = 1..infinity), 120);
MATHEMATICA
RealDigits[Chop[N[Product[(1 + 1/n^5), {n, 1, Infinity}], 120]]][[1]]
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]]]
N[1/Abs[Gamma[Exp[2*Pi*I/5]]*Gamma[Exp[6*Pi*I/5]]]^2, 120] (* Vaclav Kotesovec, Apr 27 2020 *)
PROG
(PARI) default(realprecision, 120); exp(sumalt(j=1, -(-1)^j*zeta(5*j)/j))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Mar 29 2019
STATUS
approved