A331839 -id:A331839 - cp's OEIS frontend

A013663 Decimal expansion of zeta(5).

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

Offset: 1

N. J. A. Sloane

In a widely distributed May 2011 email, Wadim Zudilin gave a rebuttal to v1 of Kim's 2011 preprint: "The mistake (unfixable) is on p. 6, line after eq. (3.3). 'Without loss of generality' can be shown to work only for a finite set of n_k's; as the n_k are sufficiently large (and N is fixed), the inequality for epsilon is false." In a May 2013 email, Zudilin extended his rebuttal to cover v2, concluding that Kim's argument "implies that at least one of zeta(2), zeta(3), zeta(4) and zeta(5) is irrational, which is trivial." - Jonathan Sondow, May 06 2013

General: zeta(2*s + 1) = (A000364(s)/A331839(s)) * Pi^(2*s + 1) * Product_{k >= 1} (A002145(k)^(2*s + 1) + 1)/(A002145(k)^(2*s + 1) - 1), for s >= 1. - Dimitris Valianatos, Apr 27 2020

			1/1^5 + 1/2^5 + 1/3^5 + 1/4^5 + 1/5^5 + 1/6^5 + 1/7^5 + ... =
1 + 1/32 + 1/243 + 1/1024 + 1/3125 + 1/7776 + 1/16807 + ... = 1.036927755143369926331365486457...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael J. Dancs and Tian-Xiao He, An Euler-type formula for zeta(2k+1), Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199.
Robert J. Harley, Zeta(3), Zeta(5), .., Zeta(99) 10000 digits (txt, 400 KB).
Yong-Cheol Kim, zeta(5) is irrational, arXiv:1105.0730 [math.CA], 2011. [Jonathan Vos Post, May 4, 2011].
Simon Plouffe, Computation of Zeta(5)
Simon Plouffe, Zeta(5), the sum(1/n**5, n=1..infinity) to 512 digits
Simon Plouffe, Other interesting computations at numberworld.org.
Zhi-Wei Sun, A curious hypergeometric series related to Riemann's zeta function, Question 486353 at MathOverflow, Jan. 21, 2025.
Chuanan Wei, Some fast convergent series for the mathematical constants zeta(4) and zeta(5), arXiv:2303.07887 [math.CO], 2023.
Wikipedia, Zeta constant
Wadim Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Russ. Math. Surv., 56 (2001), 774-776.
Index entries for constants related to zeta

Cf. A013661, A002117, A013662, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
Cf. A243264, A255323.
Cf. A023872, A023873, A248882, A255050, A255052, A057528, A260404.

RealDigits[Zeta[5], 10, 100][[1]] (* Alonso del Arte, Jan 13 2012 *)

PARI
```
zeta(5) \\ Michel Marcus, Apr 17 2016
```

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(5) = Sum_{n >= 1} 1/n^5.
zeta(5) = 2^5/(2^5 - 1)*(Sum_{n even} n^5*p(n)*p(1/n)/(n^2 - 1)^6 ), where p(n) = n^2 + 3. See A013667, A013671 and A013675. (End)
zeta(5) = Sum_{n >= 1} (A010052(n)/n^(5/2)) = Sum_{n >= 1} ((floor(sqrt(n)) - floor(sqrt(n-1)))/n^(5/2)). - Mikael Aaltonen, Feb 22 2015
zeta(5) = Product_{k>=1} 1/(1 - 1/prime(k)^5). - Vaclav Kotesovec, Apr 30 2020
From Artur Jasinski, Jun 27 2020: (Start)
zeta(5) = (-1/30)*Integral_{x=0..1} log(1-x^4)^5/x^5.
zeta(5) = (1/24)*Integral_{x=0..infinity} x^4/(exp(x)-1).
zeta(5) = (2/45)*Integral_{x=0..infinity} x^4/(exp(x)+1).
zeta(5) = (1/(1488*zeta(1/2)^5))*(-5*Pi^5*zeta(1/2)^5 + 96*zeta'(1/2)^5 - 240*zeta(1/2)*zeta'(1/2)^3*zeta''(1/2) + 120*zeta(1/2)^2*zeta'(1/2)*zeta''(1/2)^2 + 80*zeta(1/2)^2*zeta'(1/2)^2*zeta'''(1/2)- 40*zeta(1/2)^3*zeta''(1/2)*zeta'''(1/2) - 20*zeta(1/2)^3*zeta'(1/2)*zeta''''(1/2)+4*zeta(1/2)^4*zeta'''''(1/2)). (End).
From Peter Bala, Oct 29 2023: (Start)
zeta(3) = (8/45)*Integral_{x >= 1} x^3*log(x)^3*(1 + log(x))*log(1 + 1/x^x) dx =  (2/45)*Integral_{x >= 1} x^4*log(x)^4*(1 + log(x))/(1 + x^x) dx.
zeta(5) = 131/128 + 26*Sum_{n >= 1} (n^2 + 2*n + 40/39)/(n*(n + 1)*(n + 2))^5.
zeta(5) =  5162893/4976640 - 1323520*Sum_{n >= 1} (n^2 + 4*n + 56288/12925)/(n*(n + 1)*(n + 2)*(n + 3)*(n + 4))^5. Taking 10 terms of the series gives a value for zeta(5) correct to 20 decimal places.
Conjecture: for k >= 1, there exist rational numbers A(k), B(k) and c(k) such that zeta(5) = A(k) + B(k)*Sum_{n >= 1} (n^2 + 2*k*n + c(k))/(n*(n + 1)*...*(n + 2*k))^5. A similar conjecture can be made for the constant zeta(3). (End)
zeta(5) = (694/204813)*Pi^5 - Sum_{n >= 1} (6280/3251)*(1/(n^5*(exp(4*Pi*n)-1))) + Sum_{n >= 1} (296/3251)*(1/(n^5*(exp(5*Pi*n)-1))) - Sum_{n >= 1} (1073/6502)*(1/(n^5*(exp(10*Pi*n)-1))) + Sum_{n >= 1} (37/6502)*(1/(n^5*(exp(20*Pi*n)-1))). - Simon Plouffe, Jan 06 2024
From Peter Bala, Apr 27 2025: (Start)
zeta(5) = 1/5! * Integral_{x >= 0} x^5 * exp(x)/(exp(x) - 1)^2 dx = (16/15) * 1/5! * Integral_{x >= 0} x^5 * exp(x)/(exp(x) + 1)^2 dx.
zeta(5) = 1/6! * Integral_{x >= 0} x^6 * exp(x)*(exp(x) + 1)/(exp(x) - 1)^3 dx = 1/(3^3 * 5^2) * Integral_{x >= 0} x^6 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
zeta(5) = Sum_{i, j >= 1} 1/((i^4)*j*binomial(i+j, i)). More generally, zeta(n+1) = Sum_{i, j >= 1} 1/((i^n)*j*binomial(i+j, i)) for n >= 1. - Peter Bala, Aug 07 2025

A340471 Denominators of an approximation to zeta(n)/Pi^n.

Original entry on oeis.org

2, 6, 28, 90, 1488, 945, 182880, 9450, 8241408, 93555, 14856307200, 638512875, 1569400842240, 18243225, 5713142135500800, 325641566250, 1096948397364019200, 38979295480125, 6713362606110031872000, 1531329465290625, 408173030347971900211200, 13447856940643125

Offset: 1

Melchor Viso Martinez, Jan 08 2021

			1/2, 1/6, 1/28, 1/90, 5/1488, 1/945, 61/182880, 1/9450, 277/8241408, 1/93555, 50521/14856307200, 691/638512875, ...
Values are approximate for odd indices, exact for even indices:
  zeta(1) ~       1/2             zeta(2) = Pi^2/6
  zeta(3) ~    Pi^3/28            zeta(4) = Pi^4/90
  zeta(5) ~  5*Pi^5/1488          zeta(6) = Pi^6/945
  zeta(7) ~ 61*Pi^7/182880,       zeta(8) = Pi^8/9450
  ...

Melchor Viso Martinez, An expression for integer zeta approximation

Cf. A000831, A002432, A331839, A340472 (numerators).

a[k_] := Denominator[(1/(4 (1 - 2^-k) k!)
      D[\[Lambda] Tan[(\[Pi] + \[Lambda])/4], {\[Lambda],
       k}]) /. {\[Lambda] -> 0}]

a(n) = {my(t=tan(x/4 + O(x*x^n))); denominator(polcoef(x*(1 + t)/(1 - t), n)/((4-2^(2-n))))} \\ Andrew Howroyd, Jan 10 2021

a(n) = denominator of lim_{x->0} of the n-th derivative of x*tan((Pi+x)/4)/((4-2^(2-n))*n!) with respect to x.
a(2*n) = A002432(n).
From Andrew Howroyd, Jan 10 2021: (Start)
a(n) = denominator of (1/(4-2^(2-n)))*[x^n] x*(1 + tan(x/4))/(1 - tan(x/4)).
a(n) = denominator( A000831(n-1)/((n-1)!*2^n*(2^n-1)) ). (End)

A352602 a(n) = 4^n(2^(2n+1)-1)(2n)!.

Original entry on oeis.org

1, 56, 11904, 5852160, 5274501120, 7606429286400, 16070664624537600, 46802060374022553600, 179724025424120905728000, 879933863508054097526784000, 5350005543376937290448240640000, 39547255119844566012586402775040000, 349281388446657765223160470894018560000

Offset: 0

Artur Jasinski, Mar 22 2022

nonn

For n>0, PolyGamma(2*n,1/4) = -a(n)*Zeta(2*n+1) - A000816(n)*Pi^(2n+1) = -2^(2*n-1)*(A331839(n)*Zeta(2*n+1) + A000364(n)*Pi^(2n+1)).

			PolyGamma(2,1/4) = -56*zeta(3) - 2*Pi^3
PolyGamma(4,1/4) = -11904*zeta(5) - 40*Pi^5
PolyGamma(6,1/4) = -5852160*zeta(7) - 1952*Pi^7

Cf. A000302, A000364, A000816, A010050, A331839.

A352602 := proc(n)
    4^n*(2^(2*n+1)-1)*(2*n)! ;
end proc:
seq(A352602(n),n=0..30) ; # R. J. Mathar, Aug 19 2022

Table[4^n*(2^(2*n + 1) - 1)*(2*n)!, {n, 0, 12}]

a(n) = n<<=1; my(f=n!<Kevin Ryde, Mar 23 2022

a(n) = (-Pi^(2*n+1)*A000816(n) - PolyGamma(2*n,1/4))/zeta(2*n+1).
a(n) = 2^(2*n-1)*A331839(n).
D-finite with recurrence a(n) -40*n*(2*n-1)*a(n-1) +256*n*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 19 2022

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A013663 Decimal expansion of zeta(5).

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A340471 Denominators of an approximation to zeta(n)/Pi^n.

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A352602 a(n) = 4^n(2^(2n+1)-1)(2n)!.

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cp's OEIS Frontend

A013663 Decimal expansion of zeta(5).

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A340471 Denominators of an approximation to zeta(n)/Pi^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A352602 a(n) = 4^n*(2^(2*n+1)-1)*(2*n)!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A352602 a(n) = 4^n(2^(2n+1)-1)(2n)!.