OFFSET
1,4
REFERENCES
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.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..2000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. H. Bailey, J. M. Borwein and D. M. Bradley, Experimental determination of Apéry-like identities for zeta(4n+2), arXiv:math/0505270 [math.NT], 2005-2006.
Ankush Goswami, A q-analogue for Euler's ζ(6) = π^6/945, arXiv:1802.08529 [math.NT], 2018.
FORMULA
Equals Pi^6/945 = A092732/945. - Mohammad K. Azarian, Mar 03 2008
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
Definition: zeta(6) = Sum_{n >= 1} 1/n^6. - Bruno Berselli, Dec 05 2013
zeta(6) = Sum_{n >= 1} (A010052(n)/n^3). - Mikael Aaltonen, Feb 20 2015
zeta(6) = Sum_{n >= 1} (A010057(n)/n^2). - A.H.M. Smeets, Sep 19 2018
zeta(6) = Product_{k>=1} 1/(1 - 1/prime(k)^6). - Vaclav Kotesovec, May 02 2020
From Wolfdieter Lang, Sep 16 2020: (Start)
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.
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)
EXAMPLE
1.01734306198444913...
MAPLE
evalf(Pi^6/945) ; # R. J. Mathar, Oct 16 2015
MATHEMATICA
RealDigits[Zeta[6], 10, 100][[1]] (* Vincenzo Librandi, Feb 15 2015 *)
PROG
(PARI) zeta(6) \\ Michel Marcus, Feb 15 2015
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved