cp's OEIS Frontend

A013666 Decimal expansion of zeta(8).

%I A013666 #49 Apr 29 2025 04:55:30
%S A013666 1,0,0,4,0,7,7,3,5,6,1,9,7,9,4,4,3,3,9,3,7,8,6,8,5,2,3,8,5,0,8,6,5,2,
%T A013666 4,6,5,2,5,8,9,6,0,7,9,0,6,4,9,8,5,0,0,2,0,3,2,9,1,1,0,2,0,2,6,5,2,5,
%U A013666 8,2,9,5,2,5,7,4,7,4,8,8,1,4,3,9,5,2,8,7,2,3,0,3,7,2,3,7,1,9,7
%N A013666 Decimal expansion of zeta(8).
%C A013666 This sequence is also the decimal expansion of Pi^8/9450. - _Mohammad K. Azarian_, Mar 03 2008
%D A013666 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.
%H A013666 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=807&amp;Submit=Go">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A013666 zeta(8) = 2/3*2^8/(2^8 - 1)*( Sum_{n even} n^2*p(n)/(n^2 - 1)^9 ), where p(n) = 5*n^8 + 60*n^6 + 126*n^4 + 60*n^2 + 5 is a row polynomial of A091043. See A013662, A013664, A013668 and A013670. - _Peter Bala_, Dec 05 2013
%F A013666 zeta(8) = Sum_{n >= 1} (A010052(n)/n^4). - _Mikael Aaltonen_, Feb 20 2015
%F A013666 zeta(8) = Product_{k>=1} 1/(1 - 1/prime(k)^8). - _Vaclav Kotesovec_, May 02 2020
%F A013666 From _Wolfdieter Lang_, Sep 16 2020 (Start):
%F A013666 zeta(8) = (1/7!)*Integral_{0..infinity} x^7/(exp(x) - 1) dx. See Abramowitz-Stegun, 23.2.7., for s=8, p. 807. The value of the integral is 8*Pi^8/15 = 5060.54987... .
%F A013666 zeta(8) = (2^7/(127*7!))*Integral_{0..infinity} x^7/(exp(x) + 1) dx. See Abramowitz-Stegun, 23.2.8., for s=8, p. 807. The prefactor is 8/40005. The value of the integral is (127/240)*Pi^8 =  5021.014329... .(End)
%F A013666 Equals A092736/9450. - _R. J. Mathar_, Jan 07 2021
%F A013666 From  _Peter Bala_, Apr 27 2025: (Start)
%F A013666 zeta(8) = 1/8! * Integral_{x >= 0} x^8 * exp(x)/(exp(x) - 1)^2 dx = 2^7/(2^7 - 1) * 1/8! * Integral_{x >= 0} x^8 * exp(x)/(exp(x) + 1)^2 dx.
%F A013666 zeta(8) = 1/9! * Integral_{x >= 0} x^9 * exp(x)*(exp(x) + 1) /(exp(x) - 1)^3 dx = 1/(3*15*63*127) * Integral_{x >= 0} x^9 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
%e A013666 1.00407735619794433937868523850865246525896079064985002032911020265...
%p A013666 Digits := 100 : evalf(Pi^8/9450) ; # _R. J. Mathar_, Jan 07 2021
%t A013666 RealDigits[Zeta[8], 10, 100][[1]] (* _Vincenzo Librandi_, Feb 15 2015 *)
%Y A013666 Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
%K A013666 nonn,cons
%O A013666 1,4
%A A013666 _N. J. A. Sloane_