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A013663 Decimal expansion of zeta(5).

%I A013663 #178 Aug 08 2025 00:48:01
%S A013663 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,
%T A013663 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,
%U A013663 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
%N A013663 Decimal expansion of zeta(5).
%C A013663 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
%C A013663 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
%D A013663 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
%D A013663 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.
%H A013663 Milton Abramowitz and Irene A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A013663 Michael J. Dancs and Tian-Xiao He, <a href="http://dx.doi.org/10.1016/j.jnt.2005.09.005">An Euler-type formula for zeta(2k+1)</a>, Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199.
%H A013663 Robert J. Harley, <a href="http://www.plouffe.fr/simon/constants/zeta3to99.txt">Zeta(3), Zeta(5), .., Zeta(99)</a> 10000 digits (txt, 400 KB).
%H A013663 Yong-Cheol Kim, <a href="http://arxiv.org/abs/1105.0730">zeta(5) is irrational</a>, arXiv:1105.0730 [math.CA], 2011. [Jonathan Vos Post, May 4, 2011].
%H A013663 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/zeta5.txt">Computation of Zeta(5)</a>
%H A013663 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap99.html">Zeta(5), the sum(1/n**5, n=1..infinity) to 512 digits</a>
%H A013663 Simon Plouffe, <a href="http://www.numberworld.org/y-cruncher/records.html">Other interesting computations</a> at numberworld.org.
%H A013663 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/486353">A curious hypergeometric series related to Riemann's zeta function</a>, Question 486353 at MathOverflow, Jan. 21, 2025.
%H A013663 Chuanan Wei, <a href="https://arxiv.org/abs/2303.07887">Some fast convergent series for the mathematical constants zeta(4) and zeta(5)</a>, arXiv:2303.07887 [math.CO], 2023.
%H A013663 Wikipedia, <a href="http://en.wikipedia.org/wiki/Zeta_constant#Odd_positive_integers">Zeta constant</a>
%H A013663 Wadim Zudilin, <a href="http://dx.doi.org/10.4213/rm427">One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational</a>, Russ. Math. Surv., 56 (2001), 774-776.
%H A013663 <a href="/wiki/Index_to_constants#Start_of_section_Z">Index entries for constants related to zeta</a>
%F A013663 From _Peter Bala_, Dec 04 2013: (Start)
%F A013663 Definition: zeta(5) = Sum_{n >= 1} 1/n^5.
%F A013663 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)
%F A013663 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
%F A013663 zeta(5) = Product_{k>=1} 1/(1 - 1/prime(k)^5). - _Vaclav Kotesovec_, Apr 30 2020
%F A013663 From _Artur Jasinski_, Jun 27 2020: (Start)
%F A013663 zeta(5) = (-1/30)*Integral_{x=0..1} log(1-x^4)^5/x^5.
%F A013663 zeta(5) = (1/24)*Integral_{x=0..infinity} x^4/(exp(x)-1).
%F A013663 zeta(5) = (2/45)*Integral_{x=0..infinity} x^4/(exp(x)+1).
%F A013663 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).
%F A013663 From _Peter Bala_, Oct 29 2023: (Start)
%F A013663 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.
%F A013663 zeta(5) = 131/128 + 26*Sum_{n >= 1} (n^2 + 2*n + 40/39)/(n*(n + 1)*(n + 2))^5.
%F A013663 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.
%F A013663 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)
%F A013663 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
%F A013663 From _Peter Bala_, Apr 27 2025: (Start)
%F A013663 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.
%F A013663 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)
%F A013663 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
%e A013663 1/1^5 + 1/2^5 + 1/3^5 + 1/4^5 + 1/5^5 + 1/6^5 + 1/7^5 + ... =
%e A013663 1 + 1/32 + 1/243 + 1/1024 + 1/3125 + 1/7776 + 1/16807 + ... = 1.036927755143369926331365486457...
%t A013663 RealDigits[Zeta[5], 10, 100][[1]] (* _Alonso del Arte_, Jan 13 2012 *)
%o A013663 (PARI) zeta(5) \\ _Michel Marcus_, Apr 17 2016
%Y A013663 Cf. A013661, A002117, A013662, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
%Y A013663 Cf. A243264, A255323.
%Y A013663 Cf. A023872, A023873, A248882, A255050, A255052, A057528, A260404.
%K A013663 nonn,cons
%O A013663 1,3
%A A013663 _N. J. A. Sloane_