cp's OEIS Frontend

A013664 Decimal expansion of zeta(6).

%I A013664 #78 Apr 29 2025 04:55:53
%S A013664 1,0,1,7,3,4,3,0,6,1,9,8,4,4,4,9,1,3,9,7,1,4,5,1,7,9,2,9,7,9,0,9,2,0,
%T A013664 5,2,7,9,0,1,8,1,7,4,9,0,0,3,2,8,5,3,5,6,1,8,4,2,4,0,8,6,6,4,0,0,4,3,
%U A013664 3,2,1,8,2,9,0,1,9,5,7,8,9,7,8,8,2,7,7,3,9,7,7,9,3,8,5,3,5,1,7
%N A013664 Decimal expansion of zeta(6).
%D A013664 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.
%D A013664 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
%H A013664 Muniru A Asiru, <a href="/A013664/b013664.txt">Table of n, a(n) for n = 1..2000</a>
%H A013664 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].
%H A013664 D. H. Bailey, J. M. Borwein and D. M. Bradley, <a href="https://arxiv.org/abs/math/0505270">Experimental determination of Apéry-like identities for zeta(4n+2)</a>, arXiv:math/0505270 [math.NT], 2005-2006.
%H A013664 Ankush Goswami, <a href="https://arxiv.org/abs/1802.08529">A q-analogue for Euler's ζ(6) = π^6/945</a>, arXiv:1802.08529 [math.NT], 2018.
%H A013664 <a href="/wiki/Index_to_constants#Start_of_section_Z">Index entries for constants related to zeta</a>.
%F A013664 Equals Pi^6/945 = A092732/945. - _Mohammad K. Azarian_, Mar 03 2008
%F A013664 zeta(6) = 8/3*2^6/(2^6 - 1)*( Sum_{n even} n^2*p(n)/(n^2 - 1)^7 ), where p(n) = n^6 + 7*n^4 + 7*n^2 + 1 is a row polynomial of A091043. See A013662, A013666, A013668 and A013670. - _Peter Bala_, Dec 05 2013
%F A013664 Definition: zeta(6) = Sum_{n >= 1} 1/n^6. - _Bruno Berselli_, Dec 05 2013
%F A013664 zeta(6) = Sum_{n >= 1} (A010052(n)/n^3). - _Mikael Aaltonen_, Feb 20 2015
%F A013664 zeta(6) = Sum_{n >= 1} (A010057(n)/n^2). - _A.H.M. Smeets_, Sep 19 2018
%F A013664 zeta(6) = Product_{k>=1} 1/(1 - 1/prime(k)^6). - _Vaclav Kotesovec_, May 02 2020
%F A013664 From _Wolfdieter Lang_, Sep 16 2020: (Start)
%F A013664 zeta(6) = (1/5!)*Integral_{x=0..infinity} x^5/(exp(x) - 1) dx. See Abramowitz-Stegun, 23.2.7., for s=6, p. 807. See also A337710 for the value of the integral.
%F A013664 zeta(6) = (4/465)*Integral_{x=0..infinity} x^5/(exp(x) + 1) dx. See Abramowitz-Stegun, 23.2.8., for s=6, p. 807. The value of the integral is (31/252)*Pi^6 = 118.2661309... . (End)
%F A013664 From _Peter Bala_, Apr 27 2025: (Start)
%F A013664 zeta(6) = 1/6! * Integral_{x >= 0} x^6 * exp(x)/(exp(x) - 1)^2 dx = 2^5/(2^5 - 1) * 1/6! * Integral_{x >= 0} x^6 * exp(x)/(exp(x) + 1)^2 dx.
%F A013664 zeta(6) = 1/7! * Integral_{x >= 0} x^7 * exp(x)*(exp(x) + 1) /(exp(x) - 1)^3 dx = 2/(3*7*15*31) * Integral_{x >= 0} x^7 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
%e A013664 1.01734306198444913...
%p A013664 evalf(Pi^6/945) ;  # _R. J. Mathar_, Oct 16 2015
%t A013664 RealDigits[Zeta[6], 10, 100][[1]] (* _Vincenzo Librandi_, Feb 15 2015 *)
%o A013664 (PARI) zeta(6) \\ _Michel Marcus_, Feb 15 2015
%Y A013664 Cf. A013661, A002117, A013662, A013663, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
%Y A013664 Cf. A337710.
%K A013664 nonn,cons
%O A013664 1,4
%A A013664 _N. J. A. Sloane_