A175616 -id:A175616 - cp's OEIS frontend

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

Vaclav Kotesovec, Apr 27 2020

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)

			2.00815605499274531514903948232341369211953215983095097877074299617422...

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

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

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

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

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

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).

A339745 Decimal expansion of Product_{n>=2} (1 - n^(-10)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 15 2020

			0.99900544248098947527378453585422724586059097385364736908229...

Cf. A109219, A175615, A175616, A175617, A175618, A175619.

evalf((cosh(sqrt((5 - sqrt(5))/2)*Pi) + sin(sqrt(5)*Pi/2)) * (cosh(sqrt((5 + sqrt(5))/2)*Pi) - sin(sqrt(5)*Pi/2)) / (40*Pi^4), 100);

RealDigits[(Cosh[Sqrt[(5 - Sqrt[5])/2]*Pi] + Sin[Sqrt[5]*Pi/2]) * (Cosh[Sqrt[(5 + Sqrt[5])/2]*Pi] - Sin[Sqrt[5]*Pi/2]) / (40*Pi^4), 10, 100][[1]]

PARI
```
exp(suminf(j=1, (1 - zeta(10*j))/j))
```

prodinf(n=2, 1-1/n^10) \\ Michel Marcus, Dec 15 2020

Equals (cosh(sqrt((5 - sqrt(5))/2)*Pi) + sin(sqrt(5)*Pi/2)) * (cosh(sqrt((5 + sqrt(5))/2)*Pi) - sin(sqrt(5)*Pi/2)) / (40*Pi^4).

Equals exp(Sum_{j>=1} (1 - zeta(10*j))/j).

A346745 Decimal expansion of Product_{k>=2} (1 - 1/k^12).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 31 2021

			0.999753913921893256003448570641909727180...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A109219, A175615, A175616, A175617, A175618, A175619, A339745.

evalf(sinh(Pi) * cosh(Pi*sqrt(3)/2)^2 * (cosh(Pi) - cos(Pi*sqrt(3))) / (24*Pi^5), 120); # Vaclav Kotesovec, Aug 01 2021

RealDigits[Sinh[Pi]*Cosh[Pi*Sqrt[3]/2]^2*(Cosh[Pi] - Cos[Pi*Sqrt[3]])/(24*Pi^5), 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

exp(suminf(j=1, (1 - zeta(12*j))/j)) \\ Vaclav Kotesovec, Aug 01 2021

Equals sinh(Pi) * cosh(Pi*sqrt(3)/2)^2 * (cosh(Pi) - cos(Pi*sqrt(3))) / (24*Pi^5).

Equals exp(Sum_{j>=1} (1 - zeta(12*j))/j). - Vaclav Kotesovec, Aug 01 2021

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

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A339745 Decimal expansion of Product_{n>=2} (1 - n^(-10)).

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A346745 Decimal expansion of Product_{k>=2} (1 - 1/k^12).

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