A334411 Decimal expansion of Product_{k>=1} (1 + 1/k^8).

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, 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, 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

Offset

1,1

Comments

From Vaclav Kotesovec, Aug 30 2024: (Start)

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).

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)).

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)

Links

Table of n, a(n) for n=1..105.

Formula

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

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).

Example

2.00815605499274531514903948232341369211953215983095097877074299617422...

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A192058 A021502 A319568 * A028698 A013667 A091933

Adjacent sequences: A334408 A334409 A334410 * A334412 A334413 A334414

Keyword

nonn,cons

Author

Vaclav Kotesovec, Apr 27 2020

Status

approved