A013672 -id:A013672 - cp's OEIS frontend

A013662 Decimal expansion of zeta(4).

1, 0, 8, 2, 3, 2, 3, 2, 3, 3, 7, 1, 1, 1, 3, 8, 1, 9, 1, 5, 1, 6, 0, 0, 3, 6, 9, 6, 5, 4, 1, 1, 6, 7, 9, 0, 2, 7, 7, 4, 7, 5, 0, 9, 5, 1, 9, 1, 8, 7, 2, 6, 9, 0, 7, 6, 8, 2, 9, 7, 6, 2, 1, 5, 4, 4, 4, 1, 2, 0, 6, 1, 6, 1, 8, 6, 9, 6, 8, 8, 4, 6, 5, 5, 6, 9, 0, 9, 6, 3, 5, 9, 4, 1, 6, 9, 9, 9, 1

Offset: 1

N. J. A. Sloane

			1.082323233711138191516003696541167...

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.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
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.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 162.
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 values arXiv: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.
Romeo Meštrović, 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. - From _N. 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).
K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
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.

Cf. A013661, A002117, A013663, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
Cf. A231535, A337711.
Cf. A001008, A002805.
See also the extensive crossref table in A308637.

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

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

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

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

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

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

zeta(4) = Pi^4/90 = A092425/90. - Harry J. Smith, Apr 29 2009
From Peter 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 of A091043. See A013664, A013666, A013668 and A013670. - 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
From Wolfdieter 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 also A231535.
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 also A337711. (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
From Peter 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)
From Peter Bala, Apr 27 2025: (Start)
zeta(4) = 1/4! * Integral_{x >= 0} x^4 * exp(x)/(exp(x) - 1)^2 dx = 8/7 * 1/4! * Integral_{x >= 0} x^4 * exp(x)/(exp(x) + 1)^2 dx.
zeta(4) = 1/5! * Integral_{x >= 0} x^5 * exp(x)*(exp(x) + 1)/(exp(x) - 1)^3 dx = 1/(3*5*7) * Integral_{x >= 0} x^5 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
10*zeta(4) = Sum_{k>=1} H(k)^3/(k*(k+1)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Ramachandra, 1981). - Amiram Eldar, May 30 2025
zeta(4) = Integral_{x=0..1} Li(3,x)/x dx, where Li(n,x) is the polylogarithm function. - Kritsada Moomuang, Jun 14 2025
zeta(4) = Sum_{i, j >= 1} 1/(i^3*j*binomial(i+j, i)) = 4/3 * Sum_{i, j >= 1} 1/(i^2*j^2*binomial(i+j, i)). - Peter Bala, Aug 03 2025

A013663 Decimal expansion of zeta(5).

Original entry on oeis.org

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, 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, 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

Offset: 1

N. J. A. Sloane

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

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

			1/1^5 + 1/2^5 + 1/3^5 + 1/4^5 + 1/5^5 + 1/6^5 + 1/7^5 + ... =
1 + 1/32 + 1/243 + 1/1024 + 1/3125 + 1/7776 + 1/16807 + ... = 1.036927755143369926331365486457...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
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.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael J. Dancs and Tian-Xiao He, An Euler-type formula for zeta(2k+1), Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199.
Robert J. Harley, Zeta(3), Zeta(5), .., Zeta(99) 10000 digits (txt, 400 KB).
Yong-Cheol Kim, zeta(5) is irrational, arXiv:1105.0730 [math.CA], 2011. [Jonathan Vos Post, May 4, 2011].
Simon Plouffe, Computation of Zeta(5)
Simon Plouffe, Zeta(5), the sum(1/n**5, n=1..infinity) to 512 digits
Simon Plouffe, Other interesting computations at numberworld.org.
Zhi-Wei Sun, A curious hypergeometric series related to Riemann's zeta function, Question 486353 at MathOverflow, Jan. 21, 2025.
Chuanan Wei, Some fast convergent series for the mathematical constants zeta(4) and zeta(5), arXiv:2303.07887 [math.CO], 2023.
Wikipedia, Zeta constant
Wadim Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Russ. Math. Surv., 56 (2001), 774-776.
Index entries for constants related to zeta

Cf. A013661, A002117, A013662, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
Cf. A243264, A255323.
Cf. A023872, A023873, A248882, A255050, A255052, A057528, A260404.

RealDigits[Zeta[5], 10, 100][[1]] (* Alonso del Arte, Jan 13 2012 *)

PARI
```
zeta(5) \\ Michel Marcus, Apr 17 2016
```

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(5) = Sum_{n >= 1} 1/n^5.
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)
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
zeta(5) = Product_{k>=1} 1/(1 - 1/prime(k)^5). - Vaclav Kotesovec, Apr 30 2020
From Artur Jasinski, Jun 27 2020: (Start)
zeta(5) = (-1/30)*Integral_{x=0..1} log(1-x^4)^5/x^5.
zeta(5) = (1/24)*Integral_{x=0..infinity} x^4/(exp(x)-1).
zeta(5) = (2/45)*Integral_{x=0..infinity} x^4/(exp(x)+1).
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).
From Peter Bala, Oct 29 2023: (Start)
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.
zeta(5) = 131/128 + 26*Sum_{n >= 1} (n^2 + 2*n + 40/39)/(n*(n + 1)*(n + 2))^5.
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.
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)
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
From Peter Bala, Apr 27 2025: (Start)
zeta(5) = 1/5! * Integral_{x >= 0} x^5 * exp(x)/(exp(x) - 1)^2 dx = (16/15) * 1/5! * Integral_{x >= 0} x^5 * exp(x)/(exp(x) + 1)^2 dx.
zeta(5) = 1/6! * Integral_{x >= 0} x^6 * exp(x)*(exp(x) + 1)/(exp(x) - 1)^3 dx = 1/(3^3 * 5^2) * Integral_{x >= 0} x^6 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
zeta(5) = Sum_{i, j >= 1} 1/((i^4)*j*binomial(i+j, i)). More generally, zeta(n+1) = Sum_{i, j >= 1} 1/((i^n)*j*binomial(i+j, i)) for n >= 1. - Peter Bala, Aug 07 2025

A013664 Decimal expansion of zeta(6).

Original entry on oeis.org

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, 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, 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

Offset: 1

N. J. A. Sloane

			1.01734306198444913...

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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.

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.
Index entries for constants related to zeta.

Cf. A013661, A002117, A013662, A013663, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.

Cf. A337710.

evalf(Pi^6/945) ;  # R. J. Mathar, Oct 16 2015

RealDigits[Zeta[6], 10, 100][[1]] (* Vincenzo Librandi, Feb 15 2015 *)

PARI
```
zeta(6) \\ Michel Marcus, Feb 15 2015
```

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)
From Peter Bala, Apr 27 2025: (Start)
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.
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)

A013665 Decimal expansion of zeta(7).

Original entry on oeis.org

1, 0, 0, 8, 3, 4, 9, 2, 7, 7, 3, 8, 1, 9, 2, 2, 8, 2, 6, 8, 3, 9, 7, 9, 7, 5, 4, 9, 8, 4, 9, 7, 9, 6, 7, 5, 9, 5, 9, 9, 8, 6, 3, 5, 6, 0, 5, 6, 5, 2, 3, 8, 7, 0, 6, 4, 1, 7, 2, 8, 3, 1, 3, 6, 5, 7, 1, 6, 0, 1, 4, 7, 8, 3, 1, 7, 3, 5, 5, 7, 3, 5, 3, 4, 6, 0, 9, 6, 9, 6, 8, 9, 1, 3, 8, 5, 1, 3, 2

Offset: 1

N. J. A. Sloane

From Dimitris Valianatos, Apr 29 2020: (Start)
Let p_n = Product_{k >= 1, 4*k-1 is prime} (((4*k - 1)^n + 1) / ((4*k - 1)^n - 1)).
Then (2^(n + 1) / (2^n - 1)) * Sum_{k >= 1} 1 / (4*k - 3)^n = ((p_n + 1) / p_n) * Sum_{k >= 1} 1 / k^n = ((p_n + 1) / p_n) * zeta(n), n >= 3 odd number.
For n = 7, p_7 = 1.00091744947834007403796003463414...
The product (256 / 127) * Sum_{k >= 1} 1 / (4*k - 3)^7 = 2.01577429320860871987548541116538... is equal to the product ((p_7 + 1) / p_7) * Sum_{k >= 1} 1 / k^7 = 1.9990833914636834116748... * zeta(7) = 2.01577429320860871987548541116538... (End)

			1.0083492773819228268397975498497967595998635605652387064172831365716014...

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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Jakob Ablinger, Proving two conjectural series for zeta(7) and discovering more series for zeta(7), arXiv:1908.06631 [math.CO], 2019.
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].
J. Borwein and D. Bradley, Empirically determined Apéry-like formulas for zeta(4n+3), arXiv:math/0505124 [math.CA], 2005.
Michael J. Dancs and Tian-Xiao He, An Euler-type formula for zeta(2k+1), Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199.
Simon Plouffe, Plouffe's Inverter, Zeta(7) to 50000 digits.
Simon Plouffe, Zeta(7) to 512 places:sum(1/n^7, n=1..infinity).

Cf. A013661, A002117, A013662, A013663, A013664, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.

Cf. A023874, A023875, A248884.

RealDigits[Zeta[7],10,120][[1]] (* Harvey P. Dale, Oct 23 2012 *)

PARI
```
zeta(7) \\ Michel Marcus, Apr 17 2016
```

zeta(7) = Sum_{n >= 1} (A010052(n)/n^(7/2)) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^(7/2) ). - Mikael Aaltonen, Feb 22 2015
zeta(7) = Product_{k>=1} 1/(1 - 1/prime(k)^7). - Vaclav Kotesovec, Apr 30 2020
From Artur Jasinski, Jun 27 2020: (Start)
zeta(7) = (-1/840)*Integral_{x=0..1} log(1-x^6)^7/x^7.
zeta(7) = (1/720)*Integral_{x=0..oo} x^6/(exp(x)-1).
zeta(7) = (4/2835)*Integral_{x=0..oo} x^6/(exp(x)+1).
zeta(7) = (1/(182880*Zeta(1/2)^7))*(-61*Pi^7*zeta(1/2)^7 + 2880* zeta'(1/2)^7 - 10080*zeta(1/2)*zeta'(1/2)^5*zeta''(1/2) + 10080* zeta(1/2)^2*zeta'(1/2)^3*zeta''(1/2)^2 - 2520*zeta(1/2)^3*zeta'(1/2)* zeta''(1/2)^3 + 3360*zeta(1/2)^2*zeta'(1/2)^4*zeta'''(1/2) - 5040 zeta(1/2)^3*zeta'(1/2)^2*zeta''(1/2)*zeta'''(1/2) + 840*zeta(1/2)^4* zeta''(1/2)^2*zeta'''(1/2) + 560*zeta(1/2)^4*zeta'(1/2)*zeta'''(1/2)^3  - 840*zeta(1/2)^3*zeta'(1/2)^3*zeta''''(1/2) + 840*zeta(1/2)^4*zeta'(1/2)* zeta''(1/2)*zeta''''(1/2) - 140*zeta(1/2)^5*zeta'''(1/2)*zeta''''(1/2) + 168*zeta(1/2)^4*zeta'(1/2)^2*zeta'''''(1/2) - 84*zeta(1/2)^5*zeta''(1/2)* zeta'''''(1/2) - 28*zeta(1/2)^5*zeta'(1/2)*zeta''''''(1/2) + 4* zeta(1/2)^6*zeta'''''''(1/2)). (End)
Equals 19*Pi^7/56700 - 2*Sum_{k>=1} 1/(k^7*(exp(2*Pi*k) - 1)) [Grosswald] (see Finch). - Stefano Spezia, Nov 01 2024
From Peter Bala, Apr 27 2025: (Start)
zeta(7) = 1/7! * Integral_{x >= 0} x^7 * exp(x)/(exp(x) - 1)^2 dx = 2^6/(2^6 - 1) * 1/7! * Integral_{x >= 0} x^7 * exp(x)/(exp(x) + 1)^2 dx.
zeta(7) = 1/8! * Integral_{x >= 0} x^8 * exp(x)*(exp(x) + 1) /(exp(x) - 1)^3 dx = 1/ (2*3*7*15*63) * Integral_{x >= 0} x^8 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)

A013667 Decimal expansion of zeta(9).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 8, 3, 9, 2, 8, 2, 6, 0, 8, 2, 2, 1, 4, 4, 1, 7, 8, 5, 2, 7, 6, 9, 2, 3, 2, 4, 1, 2, 0, 6, 0, 4, 8, 5, 6, 0, 5, 8, 5, 1, 3, 9, 4, 8, 8, 8, 7, 5, 6, 5, 4, 8, 5, 9, 6, 6, 1, 5, 9, 0, 9, 7, 8, 5, 0, 5, 3, 3, 9, 0, 2, 5, 8, 3, 9, 8, 9, 5, 0, 3, 9, 3, 0, 6, 9, 1, 2, 7, 1, 6, 9, 5, 8

Offset: 1

N. J. A. Sloane

			1.0020083928260822...

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.

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].
Simon Plouffe, Plouffe's Inverter, Zeta(9)=sum(1/n^9, n=1..infinity); to 20000 digits
Simon Plouffe, Zeta(9) or sum(1/n**9, n=1..infinity);

Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013666, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.

Cf. A023876, A023877.

evalf(Zeta(9)) ; # R. J. Mathar, Oct 16 2015

RealDigits[Zeta[9],10,100][[1]] (* Harvey P. Dale, Aug 27 2014 *)

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(9) = Sum_{n >= 1} 1/n^9.
zeta(9) = 2^9/(2^9 - 1)*( Sum_{n even} n^7*p(n)*p(1/n)/(n^2 - 1)^10 ), where p(n) = n^4 + 10*n^2 + 5. See A013663, A013671 and A013675. (End)
zeta(9) = Sum_{n >= 1} (A010052(n)/n^(9/2)) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^(9/2) ). - Mikael Aaltonen, Feb 22 2015
zeta(9) = Product_{k>=1} 1/(1 - 1/prime(k)^9). - Vaclav Kotesovec, May 02 2020
From  Peter Bala, Apr 27 2025: (Start)
zeta(9) = 1/9! * Integral_{x >= 0} x^9 * exp(x)/(exp(x) - 1)^2 dx = 2^9/(2^9 - 1) * 1/9! * Integral_{x >= 0} x^9 * exp(x)/(exp(x) + 1)^2 dx.
zeta(9) = 1/10! * Integral_{x >= 0} x^10 * exp(x)*(exp(x) + 1)/(exp(x) - 1)^3 dx = 1/(3^5 * 5^3 * 7 * 17) * Integral_{x >= 0} x^10 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)

A013666 Decimal expansion of zeta(8).

Original entry on oeis.org

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, 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, 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

Offset: 1

N. J. A. Sloane

This sequence is also the decimal expansion of Pi^8/9450. - Mohammad K. Azarian, Mar 03 2008

			1.00407735619794433937868523850865246525896079064985002032911020265...

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.

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].

Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.

Digits := 100 : evalf(Pi^8/9450) ; # R. J. Mathar, Jan 07 2021

RealDigits[Zeta[8], 10, 100][[1]] (* Vincenzo Librandi, Feb 15 2015 *)

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
zeta(8) = Sum_{n >= 1} (A010052(n)/n^4). - Mikael Aaltonen, Feb 20 2015
zeta(8) = Product_{k>=1} 1/(1 - 1/prime(k)^8). - Vaclav Kotesovec, May 02 2020
From Wolfdieter Lang, Sep 16 2020 (Start):
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... .
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)
Equals A092736/9450. - R. J. Mathar, Jan 07 2021
From  Peter Bala, Apr 27 2025: (Start)
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.
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)

A092759 a(n) = prime(n)^7.

Original entry on oeis.org

128, 2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, 17249876309, 27512614111, 94931877133, 194754273881, 271818611107, 506623120463, 1174711139837, 2488651484819, 3142742836021, 6060711605323

Offset: 1

Jorge Coveiro, Apr 13 2004

Seventh powers of prime numbers. - Wesley Ivan Hurt, Mar 27 2014

			a(1) = 128 since the seventh power of the first prime is 2^7 = 128. - _Wesley Ivan Hurt_, Mar 27 2014

Subsequence of A030626.

Cf. A085967, A013665, A013672.

[p^7: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A092759:=n->ithprime(n)^7; seq(A092759(n), n=1..30); # Wesley Ivan Hurt, Mar 27 2014

Array[Prime[ # ]^7 &, 30] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Table[Prime[n]^7, {n, 30}] (* Wesley Ivan Hurt, Mar 27 2014 *)

a(n)=prime(n)^7 \\ Charles R Greathouse IV, Jul 20 2011

a(n) = A086874(n-1), n>1. - R. J. Mathar, Sep 08 2008
a(n) = A000040(n)^7 = A001015(A000040(n)). - Wesley Ivan Hurt, Mar 27 2014
Sum_{n>=1} 1/a(n) = P(7) = 0.0082838328... (A085967). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 24 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(7)/zeta(14) = A013665/A013672.
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(7) = 1/A013665. (End)

A266560 Decimal expansion of the generalized Glaisher-Kinkelin constant A(14).

Original entry on oeis.org

1, 3, 3, 8, 6, 4, 4, 7, 5, 4, 2, 4, 1, 5, 3, 6, 2, 9, 9, 5, 5, 8, 0, 4, 6, 9, 5, 8, 8, 7, 3, 2, 5, 5, 1, 4, 2, 5, 4, 2, 0, 9, 2, 5, 3, 7, 0, 6, 2, 7, 4, 2, 4, 8, 0, 2, 3, 4, 0, 6, 2, 0, 9, 4, 5, 8, 9, 7, 9, 5, 3, 1, 5, 2, 8, 5, 1, 9, 6, 4, 8, 4, 5, 5, 2, 4, 5, 2, 9, 3, 1, 3, 9, 8, 7

Offset: 1

G. C. Greubel, Dec 31 2015

Also known as the 14th Bendersky constant.

			1.338644754241536299558046958873255142542092537062742480234...

G. C. Greubel, Table of n, a(n) for n = 1..2002

Cf. A019727 (A(0)), A074962 (A(1)), A243262 (A(2)), A243263 (A(3)), A243264 (A(4)), A243265 (A(5)), A266553 (A(6)), A266554 (A(7)), A266555 (A(8)), A266556 (A(9)), A266557 (A(10)), A266558 (A(11)), A266559 (A(12)), A260662 (A(13)), A266562 (A(15)), A266563 (A(16)), A266564 (A(17)), A266565 (A(18)), A266566 (A(19)), A266567 (A(20)).

Cf. A013672, A013673, A266264, A027641, A027642.

Exp[N[(BernoulliB[14]/4)*(Zeta[15]/Zeta[14]), 200]]

A(k) = exp(H(k)*B(k+1)/(k+1) - zeta'(-k)), where B(k) is the k-th Bernoulli number, H(k) the k-th Harmonic number, and zeta'(x) is the derivative of the Riemann zeta function.
A(14) = exp(-zeta'(-14)) = exp((B(14)/4)*(zeta(15)/zeta(14))).
A(14) = exp(14! * Zeta(15) / (2^15 * Pi^14)). - Vaclav Kotesovec, Jan 01 2016

A013961 a(n) = sigma_13(n), the sum of the 13th powers of the divisors of n.

Original entry on oeis.org

1, 8193, 1594324, 67117057, 1220703126, 13062296532, 96889010408, 549822930945, 2541867422653, 10001220711318, 34522712143932, 107006334784468, 302875106592254, 793811662272744, 1946196290656824, 4504149450301441, 9904578032905938, 20825519793796029, 42052983462257060

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_{p prime} ((p^((e(p)+1)*k)) - 1)/(p^k - 1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
By Fermat's little theorem n^13 == n (mod 13). Hence sigma_13(n) == sigma_(1) (mod 13). In fact, sigma_13(n) == sigma_(1) (mod 2730), where 2730 = 2*3*5*7*13 = the numerator of Bernoulli(12). - Peter Bala, Jan 12 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n).

Cf. A000203, A001157-A001160, A013672, A013954-A013972, A017665-A017712.

[DivisorSigma(13, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016

A013961 := proc(n)
    numtheory[sigma][13](n) ;
end proc: # R. J. Mathar, Sep 21 2017

DivisorSigma[13, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *)

my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^13*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016

a(n) = sigma(n, 13); \\ Michel Marcus, Sep 10 2016

[sigma(n,13)for n in range(1,16)] # Zerinvary Lajos, Jun 04 2009

G.f.: Sum_{k>=1} k^13*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-13)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24. - Simon Plouffe, Mar 01 2021
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(13*e+13)-1)/(p^13-1).
Sum_{k=1..n} a(k) = zeta(14) * n^14 / 14 + O(n^15). (End)

A293904 Decimal expansion of zeta(21).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 4, 7, 6, 9, 3, 2, 9, 8, 6, 7, 8, 7, 8, 0, 6, 4, 6, 3, 1, 1, 6, 7, 1, 9, 6, 0, 4, 3, 7, 3, 0, 4, 5, 9, 6, 6, 4, 4, 6, 6, 9, 4, 7, 8, 4, 9, 3, 7, 6, 0, 0, 2, 0, 7, 4, 8, 7, 3, 7, 6, 5, 9, 6, 8, 3, 9, 0, 8, 7, 8, 9, 8, 1, 5, 9, 8, 3, 3, 8, 7, 6, 6

Offset: 1

Frank Ellermann, Oct 19 2017

Web searches find 1.0000004769329867878 in Python tools. Simon Plouffe published 1000 digits for zeta(9) up to zeta(2051) many years ago.

			1.000000476932986787806...

Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each.
Simon Plouffe, Zeta(9) to Zeta(2051) in reverse order to 1000 digits each. [Cached copy, with permission]
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each. (gzipped file)
Simon Plouffe, Identities inspired from Ramanujan Notebooks II, July 21, 1998.

Cf. A013661, A002117, A013662, A013663, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678.

RealDigits[Zeta[21], 10, 100][[1]] (* Amiram Eldar, May 31 2021 *)

cp's OEIS Frontend

A013662 Decimal expansion of zeta(4).

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A013663 Decimal expansion of zeta(5).

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A013664 Decimal expansion of zeta(6).

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A013665 Decimal expansion of zeta(7).

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A013667 Decimal expansion of zeta(9).

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A013666 Decimal expansion of zeta(8).

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