1,3

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F17, Series associated with the zeta-function, p. 391.

L. D. Landau and E. M. Lifschitz, Band V, Statistische Physik, Akademie Verlag, 1966, pp. 172 and 180-181.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 33.

Harry J. Smith,Table of n, a(n) for n = 1..20000

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].

Peter Bala,New series for old functions.

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.

D. Borwein and J. M. Borwein,On an intriguing integral and some series related to zeta(4)Proc. Amer. Math. Soc., Vol. 123, No.4, April 1995.

J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer,Central binomial sums, multiple Clausen values and zeta valuesarXiv:hep-th/0004153, 2000.

Leonhard Euler,On the sums of series of reciprocals,arXiv:math/0506415 [math.HO], 2005-2008.

Leonhard Euler,De summis serierum reciprocarum,E41.

Raffaele Marcovecchio and Wadim Zudilin,Hypergeometric rational approximations to zeta(4),arXiv:1905.12579 [math.NT], 2019.

R. Mestrovic,Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof,arXiv:1202.3670 [math.HO], 2012. - FromN. J. A. Sloane,Jun 13 2012

Jean-Christophe Pain,An integral representation for zeta(4),arXiv:2309.00539 [math.NT], 2023.

Michael Penn,Finding a closed form for zeta(4),YouTube video, 2022.

Simon Plouffe,Pi^4/90 to 100000 digits.

Simon Plouffe,Zeta(4) or Pi^4/90 to 10000 places.

Simon Plouffe,Zeta(2) to Zeta(4096) to 2048 digits each(gzipped file).

Carsten Schneider and Wadim Zudilin,A case study for zeta(4),arXiv:2004.08158 [math.NT], 2020.

Chuanan Wei,Some fast convergent series for the mathematical constants zeta(4) and zeta(5),arXiv:2303.07887 [math.CO], 2023.

Index entries for transcendental numbers.

zeta(4) = Pi^4/90. -Harry J. Smith,Apr 29 2009

FromPeter Bala,Dec 03 2013: (Start)

Definition: zeta(4):= Sum_{n >= 1} 1/n^4.

zeta(4) = (4/17)*Sum_{n >= 1} ( (1 + 1/2 +... + 1/n)/n )^2 and

zeta(4) = (16/45)*Sum_{n >= 1} ( (1 + 1/3 +... + 1/(2*n-1))/n )^2 (see Borwein and Borwein).

zeta(4) = (256/90)*Sum_{n >= 1} n^2*(4*n^2 + 3)*(12*n^2 + 1)/(4*n^2 - 1)^5.

Series acceleration formulas:

zeta(4) = (36/17)*Sum_{n >= 1} 1/( n^4*binomial(2*n,n) ) (Comtet)

= (36/17)*Sum_{n >= 1} P(n)/( (2*n*(2*n - 1))^4*binomial(4*n,2*n) )

= (36/17)*Sum_{n >= 1} Q(n)/( (3*n*(3*n - 1)*(3*n - 2))^4*binomial(6*n,3*n) ),

where P(n) = 80*n^4 - 48*n^3 + 24*n^2 - 8*n + 1 and Q(n) = 137781*n^8 - 275562*n^7 + 240570*n^6 - 122472*n^5 + 41877*n^4 - 10908*n^3 + 2232*n^2 - 288*n + 16 (see section 8 in the Bala link). (End)

zeta(4) = 2/3*2^4/(2^4 - 1)*( Sum_{n even} n^2*p(n)/(n^2 - 1)^5 ), where p(n) = 3*n^4 + 10*n^2 + 3 is a row polynomial ofA091043.SeeA013664,A013666,A013668andA013670.-Peter Bala,Dec 05 2013

zeta(4) = Sum_{n >= 1} ((floor(sqrt(n))-floor(sqrt(n-1)))/n^2). -Mikael Aaltonen,Jan 18 2015

zeta(4) = Product_{k>=1} 1/(1 - 1/prime(k)^4). -Vaclav Kotesovec,May 02 2020

FromWolfdieter Lang,Sep 16 2020: (Start)

zeta(4) = (1/3!)*Integral_{x=0..oo} x^3/(exp(x) - 1) dx. See Abramowitz-Stegun, 23.2.7., for s=2, p. 807, and Landau-Lifschitz, Band V, p. 172, eq. (4), for x=4. See alsoA231535.

zeta(4) = (4/21)*Integral_{x=0..oo} x^3/(exp(x) + 1) dx. See Abramowitz-Stegun, 23.2.8., for s=2, p. 807, and Landau-Lifschitz, Band V, p. 172, eq. (1), for x=4. See alsoA337711.(End)

zeta(4) = (72/17) * Integral_{x=0..Pi/3} x*(log(2*sin(x/2)))^2. See Richard K. Guy reference. -Bernard Schott,Jul 20 2022

FromPeter Bala,Nov 12 2023: (Start)

zeta(4) = 1 + (4/3)*Sum_{k >= 1} (1 - 2*(-1)^k)/(k*(k + 1)^4*(k + 2)) = 35053/32400 + 48*Sum_{k >= 1} (1 - 2*(-1)^k)/(k*(k + 1)*(k + 2)*(k + 3)^4*(k + 4)*(k + 5)*(k + 6)).

More generally, it appears that for n >= 0, zeta(4) = c(n) + (4/3)*(2*n + 1)!^2 * Sum_{k >= 1} (1 - 2*(-1)^k)/( (k + 2*n + 1)^3*Product_{i = 0..4*n+2} (k + i) ), where {c(n): n >= 0} is a sequence of rational approximations to zeta(4) beginning [1, 35053/32400, 2061943067/ 1905120000, 18594731931460103/ 17180389306080000, 257946156103293544441/ 238326360453941760000,...]. (End)

1.082323233711138191516003696541167...

evalf(Pi^4/90, 120); #Muniru A Asiru,Sep 19 2018

RealDigits[Zeta[4], 10, 120][[1]] (*Harvey P. Dale,Dec 18 2012 *)

(PARI) default(realprecision, 20080); x=Pi^4/90; for (n=1, 20000, d=floor(x); x=(x-d)*10; write( "b013662.txt", n, "", d)); \\Harry J. Smith,Apr 29 2009

(Maxima) ev(zeta(4), numer); /*R. J. Mathar,Feb 27 2012 */

(Magma) SetDefaultRealField(RealField(110)); L:=RiemannZeta(); Evaluate(L, 4); //G. C. Greubel,May 30 2019

(Sage) numerical_approx(zeta(4), digits=100) #G. C. Greubel,May 30 2019

Cf.A002117,A013661,A013664,A013666,A013668,A013670,A231535,A337711.

See also the extensive crossref table inA308637.

Sequence in context:A086058A241017A114314*A291362A319090A222225

Adjacent sequences:A013659A013660A013661*A013663A013664A013665

nonn,cons

approved