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%I #158 Mar 19 2024 07:04:04
%S 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 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 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 Decimal expansion of zeta(5).
%C 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 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 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 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 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 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 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 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/zeta5.txt">Computation of Zeta(5)</a>
%H 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 Simon Plouffe, <a href="http://www.numberworld.org/y-cruncher/records.html">Other interesting computations</a> at numberworld.org.
%H 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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Zeta_constant#Odd_positive_integers">Zeta constant</a>
%H 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 <a href="/wiki/Index_to_constants#Start_of_section_Z">Index entries for constants related to zeta</a>
%F From _Peter Bala_, Dec 04 2013: (Start)
%F Definition: zeta(5) = Sum_{n >= 1} 1/n^5.
%F 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 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 zeta(5) = Product_{k>=1} 1/(1 - 1/prime(k)^5). - _Vaclav Kotesovec_, Apr 30 2020
%F From _Artur Jasinski_, Jun 27 2020: (Start)
%F zeta(5) = (-1/30)*Integral_{x=0..1} log(1-x^4)^5/x^5.
%F zeta(5) = (1/24)*Integral_{x=0..infinity} x^4/(exp(x)-1).
%F zeta(5) = (2/45)*Integral_{x=0..infinity} x^4/(exp(x)+1).
%F 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 From _Peter Bala_, Oct 29 2023: (Start)
%F 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 zeta(5) = 131/128 + 26*Sum_{n >= 1} (n^2 + 2*n + 40/39)/(n*(n + 1)*(n + 2))^5.
%F 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 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 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
%e 1/1^5 + 1/2^5 + 1/3^5 + 1/4^5 + 1/5^5 + 1/6^5 + 1/7^5 + ... =
%e 1 + 1/32 + 1/243 + 1/1024 + 1/3125 + 1/7776 + 1/16807 + ... = 1.036927755143369926331365486457...
%t RealDigits[Zeta[5], 10, 100][[1]] (* _Alonso del Arte_, Jan 13 2012 *)
%o (PARI) zeta(5) \\ _Michel Marcus_, Apr 17 2016
%Y Cf. A002117, A013667, A013669, A013671, A013675, A013677, A243264, A255323.
%Y Cf. A023872, A023873, A248882, A255050, A255052, A057528, A260404.
%K nonn,cons
%O 1,3
%A _N. J. A. Sloane_
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