A002117 Decimal expansion of zeta(3) = Sum_{m >= 1} 1/m^3. (Formerly M0020)

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

Offset

1,2

Comments

Sometimes called Apéry's constant.

"A natural question is whether Zeta(3) is a rational multiple of Pi^3. This is not known, though in 1978 R. Apéry succeeded in proving that Zeta(3) is irrational. In Chapter 8 we pointed out that the probability that two random integers are relatively prime is 6/Pi^2, which is 1/Zeta(2). This generalizes to: The probability that k random integers are relatively prime is 1/Zeta(k) ... ." [Stan Wagon]

In 2001 Tanguy Rivoal showed that there are infinitely many odd (positive) integers at which zeta is irrational, including at least one value j in the range 5 <= j <= 21 (refined the same year by Zudilin to 5 <= j <= 11), at which zeta(j) is irrational. See the Rivoal link for further information and references.

The reciprocal of this constant is the probability that three integers chosen randomly using uniform distribution are relatively prime. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 13 2005

Also the value of zeta(1,2), the double zeta-function of arguments 1 and 2. - R. J. Mathar, Oct 10 2011

Also the length of minimal spanning tree for large complete graph with uniform random edge lengths between 0 and 1, cf. link to John Baez's comment. - M. F. Hasler, Sep 26 2017

Sum of the inverses of the cubes (A000578). - Michael B. Porter, Nov 27 2017

This number is the average value of sigma_2(n)/n^2 where sigma_2(n) is the sum of the squares of the divisors of n. - Dimitri Papadopoulos, Jan 07 2022

References

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 40-53.

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 84.

R. William Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics, Computers in Mathematics (Stanford CA, 1986); Lecture Notes in Pure and Appl. Math., Dekker, New York, 125 (1990), 261-284; MR 91h:11154.

Xavier Gourdon, Analyse, Les Maths en tête, Ellipses, 1994, Exemple 3, page 224.

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

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press; 6 edition (2008), pp. 47, 268-269.

A. A. Markoff, Mémoire sur la transformation de séries peu convergentes en séries très convergentes, Mém. de l'Acad. Imp. Sci. de St. Pétersbourg, XXXVII, 1890.

Paul J. Nahin, In Pursuit of Zeta-3, Princeton University Press, 2021.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stan Wagon, Mathematica In Action, W. H. Freeman and Company, NY, 1991, page 354.

A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Dover (1987), Ex. 92-93.

Links

Harry J. Smith, Table of n, a(n) for n = 1..20002

T. Amdeberhan, Faster and Faster convergent series for zeta(3), arXiv:math/9804126 [math.CO], 1998.

Kunihiro Aoki and Ryo Furue, A model for the size distribution of marine microplastics: a statistical mechanics approach, arXiv:2103.10221 [physics.ao-ph], 2021.

Peter Bala, New series for old functions.

John Baez, Comments about zeta(3), Azimuth Project blog, August 2017.

J. Borwein and D. Bradley, Empirically determined Apéry-like formulas for zeta(4n+3), arXiv:math/0505124 [math.CA], 2005.

Mainendra Kumar Dewangan and Subhra Datta, Effective permeability tensor of confined flows with wall grooves of arbitrary shape, J. of Fluid Mechanics (2020) Vol. 891.

Dr. Math, Probability of Random Numbers Being Coprime.

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

L. Euler, De summis serierum reciprocarum, E41.

X. Gourdon and P. Sebah, The Apery's constant: zeta(3)

Brady Haran and Tony Padilla, Apéry's constant (calculated with Twitter), Numberphile video (2017).

W. Janous, Around Apéry's constant, J. Inequ. Pure Appl. Math. 7(1) (2006), #35.

Yasuyuki Kachi and Pavlos Tzermias, Infinite products involving zeta(3) and Catalan's constant, Journal of Integer Sequences, 15 (2012), #12.9.4.

Masato Kobayashi, Integral representations for zeta(3) with the inverse sine function, arXiv:2108.01247 [math.NT], 2021.

M. Kondratiewa and S. Sadov, Markov's transformation of series and the WZ method, arXiv:math/0405592 [math.CA], 2004.

Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.

John Landen, Mathematical memoirs respecting a variety of subjects Vol. I, London, 1780.

F. M. S. Lima, Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study, arXiv:0910.2684 [math.NT], 2009-2012.

C. Lupu and D. Orr, Series representations for the Apéry constant zeta(3) involving the values zeta(2n), Ramanujan J. 48(3) (2019), 477-494.

R. J. Mathar, Yet another table of integrals, arXiv:1207.5845 [math.CA], 2012-2014.

G. P. Michon, Roger Apéry, Numericana.

S. D. Miller, An Easier Way to Show zeta(3) is Irrational.

Simon Plouffe, Zeta(3) or Apéry's constant to 2000 places.

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

A. van der Poorten, A Proof that Euler Missed.

Tanguy Rivoal, Irrationality of the zeta Function on Odd Integers [ps file].

Tanguy Rivoal, Irrationality of the zeta Function on Odd Integers [pdf file].

Ernst E. Scheufens, From Fourier series to rapidly convergent series for zeta(3), Mathematics Magazine, Vol. 84, No. 1 (2011), pp. 26-32.

G. Villemin's Almanach of Numbers, Constante d'Apéry (in French).

S. Wedeniwski, The value of zeta(3) to 1000000 places [Gutenberg Project Etext].

S. Wedeniwski, Plouffe's Inverter, Apery's constant to 128000026 decimal digits.

S. Wedeniwski, The value of zeta(3) to 1000000 decimal digits.

Eric Weisstein's World of Mathematics, Apéry's Constant.

Eric Weisstein's World of Mathematics, Relatively Prime.

Wikipedia, Riemann zeta function.

H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.

J. W. Wrench, Jr., Letter to N. J. A. Sloane, Feb 04 1971

Wenzhe Yang, Apéry's irrationality proof, mirror symmetry and Beukers' modular forms, arXiv:1911.02608 [math.NT], 2019.

Wadim Zudilin, An elementary proof of Apéry's theorem, arXiv:math/0202159 [math.NT], 2002.

Index entries for zeta function.

Formula

Lima gives an approximation to zeta(3) as (236*log(2)^3)/197 - 283/394*Pi*log(2)^2 + 11/394*Pi^2*log(2) + 209/394*log(sqrt(2) + 1)^3 - 5/197 + (93*Catalan*Pi)/197. - Jonathan Vos Post, Oct 14 2009 [Corrected by Wouter Meeussen, Apr 04 2010]

zeta(3) = 5/2*Integral_(x=0..2*log((1+sqrt(5))/2), x^2/(exp(x)-1)) + 10/3*(log((1+sqrt(5))/2))^3. - Seiichi Kirikami, Fri Aug 12 2011

zeta(3) = -4/3*Integral_{x=0..1} log(x)/x*log(1+x) = Integral_{x=0..1} log(x)/x*log(1-x) = -4/7*Integral_{x=0..1} log(x)/x*log((1+x)/(1-x)) = 4*Integral_{x=0..1} 1/x*log(1+x)^2 = 1/2*Integral_{x=0..1} 1/x*log(1-x)^2 = -16/7*Integral_{x=0..Pi/2} x*log(2*cos(x)) = -4/Pi*Integral_{x=0..Pi/2} x^2*log(2*cos(x)). - Jean-François Alcover, Apr 02 2013, after R. J. Mathar

From Peter Bala, Dec 04 2013: (Start)

zeta(3) = (16/7)*Sum_{k even} (k^3 + k^5)/(k^2 - 1)^4.

zeta(3) - 1 = Sum_{k >= 1} 1/(k^3 + 4*k^7) = 1/(5 - 1^6/(21 - 2^6/(55 - 3^6/(119 - ... - (n - 1)^6/((2*n - 1)*(n^2 - n + 5) - ...))))) (continued fraction).

More generally, there is a sequence of polynomials P(n,x) (of degree 2*n) such that

zeta(3) - Sum_{k = 1..n} 1/k^3 = Sum_{k >= 1} 1/( k^3*P(n,k-1)*P(n,k) ) = 1/((2*n^2 + 2*n + 1) - 1^6/(3*(2*n^2 + 2*n + 3) - 2^6/(5*(2*n^2 + 2*n + 7) - 3^6/(7*(2*n^2 + 2*n + 13) - ...)))) (continued fraction). See A143003 and A143007 for details.

Series acceleration formulas:

zeta(3) = (5/2)*Sum_{n >= 1} (-1)^(n+1)/( n^3*binomial(2*n,n) )

= (5/2)*Sum_{n >= 1} P(n)/( (2*n(2*n - 1))^3*binomial(4*n,2*n) )

= (5/2)*Sum_{n >= 1} (-1)^(n+1)*Q(n)/( (3*n(3*n - 1)*(3*n - 2))^3*binomial(6*n,3*n) ), where P(n) = 24*n^3 + 4*n^2 - 6*n + 1 and Q(n) = 9477*n^6 - 11421*n^5 + 5265*n^4 - 1701*n^3 + 558*n^2 - 108*n + 8 (Bala, section 7). (End)

zeta(3) = Sum_{n >= 1} (A010052(n)/n^(3/2)) = Sum_{n >= 1} ( (floor(sqrt(n)) - floor(sqrt(n-1)))/n^(3/2) ). - Mikael Aaltonen, Feb 22 2015

zeta(3) = Product_{k>=1} 1/(1 - 1/prime(k)^3). - Vaclav Kotesovec, Apr 30 2020

zeta(3) = 4*(2*log(2) - 1 - 2*Sum_{k>=2} zeta(2*k+1)/2^(2*k+1)). - Jorge Coveiro, Jun 21 2020

zeta(3) = (4*zeta'''(1/2)*(zeta(1/2))^2-12*zeta(1/2)*zeta'(1/2)*zeta''(1/2)+8*(zeta'(1/2))^3-Pi^3*(zeta(1/2))^3)/(28*(zeta(1/2))^3). - Artur Jasinski, Jun 27 2020

zeta(3) = Sum_{k>=1} H(k)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jul 31 2020

From Artur Jasinski, Sep 30 2020: (Start)

zeta(3) = (5/4)*Li_3(1/f^2) + Pi^2*log(f)/6 - 5*log(f)^3/6,

zeta(3) = (8/7)*Li_3(1/2) + (2/21)*Pi^2 log(2) - (4/21) log(2)^3, where f is golden ratio (A001622) and Li_3 is the polylogarithm function, formulas published by John Landen in 1780, p. 118. (End)

zeta(3) = (1/2)*Integral_{x=0..oo} x^2/(e^x-1) dx (Gourdon). - Bernard Schott, Apr 28 2021

From Peter Bala, Jan 18 2022: (Start)

zeta(3) = 1 + Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)) = 25/24 + (2!)^4*Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*(4*n^4 + 2^4)) = 28333/27000 + (3!)^4*(Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*(4*n^4 + 2^4)*(4*n^4 + 3^4)). In general, for k >= 1, we have zeta(3) = r(k) + (k!)^4*Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*...*(4*n^4 + k^4)), where r(k) is rational.

zeta(3) = (6/7) + (64/7)*Sum_{n >= 1} n/(4*n^2 - 1)^3.

More generally, for k >= 0, it appears that zeta(3) = a(k) + b(k)*Sum_{n >= 1} n/( (4*n^2 - 1)*(4*n^2 - 9)*...*(4*n^2 - (2*k+1)^2) )^3, where a(k) and b(k) are rational.

zeta(3) = (10/7) - (128/7)*Sum_{n >= 1} n/(4*n^2 - 1)^4.

More generally, for k >= 0, it appears that zeta(3) = c(k) + d(k)*Sum_{n >= 1} n/( (4*n^2 - 1)*(4*n^2 - 9)*...*(4*n^2 - (2*k+1)^2) )^4, where c(k) and d(k) are rational.

zeta(3) = 2/3 + (2^13)/(3*7)*Sum_{n >= 1} n^3/(4*n^2 - 1)^6. (End)

zeta(3) = -Psi(2)(1/2)/14 (the second derivative of digamma function evaluated at 1/2). - Artur Jasinski, Mar 18 2022

zeta(3) = -(8*Pi^2/9) * Sum_{k>=0} zeta(2*k)/((2*k+1)*(2*k+3)*4^k) = (2*Pi^2/9) * (log(2) + 2 * Sum_{k>=0} zeta(2*k)/((2*k+3)*4^k)) (Scheufens, 2011). - Amiram Eldar, May 28 2022

zeta(3) = Sum_{k>=1} (30*k-11) / (4*(2k-1)*k^3*(binomial(2k,k))^2) (Gosper, 1986 and Richard K. Guy reference). - Bernard Schott, Jul 20 2022

Example

1.2020569031595942853997...

Maple

# Calculates an approximation with n exact decimal places (small deviation
# in the last digits are possible). Goes back to ideas of A. A. Markoff 1890.
zeta3 := proc(n) local s, w, v, k; s := 0; w := -1; v := 4;
for k from 2 by 2 to 7*n/2 do
    w := -w*v/k;
    v := v + 8;
    s := s + 1/(w*k^3);
od; 20*s; evalf(%, n) end:
zeta3(10000); # Peter Luschny, Jun 10 2020

Mathematica

RealDigits[ N[ Zeta[3], 100] ] [ [1] ]
(* Second program (historical interest): *)
d[n_] := 34*n^3 + 51*n^2 + 27*n + 5; 6/Fold[Function[d[#2-1] - #2^6/#1], 5, Reverse[Range[100]]] // N[#, 108]& // RealDigits // First
(* Jean-François Alcover, Sep 19 2014, after Apéry's continued fraction *)

Prog

(PARI) default(realprecision, 20080); x=zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002117.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009
(Maxima) fpprec : 100$ ev(bfloat(zeta(3)))$ bfloat(%); /* Martin Ettl, Oct 21 2012 */
(Python)
from mpmath import mp, apery
mp.dps=109
print([int(z) for z in list(str(apery).replace('.', ''))[:-1]]) # Indranil Ghosh, Jul 08 2017
(Magma) L:=RiemannZeta(: Precision:=100); Evaluate(L, 3); // G. C. Greubel, Aug 21 2018

Crossrefs

Cf. A013631, A013679, A013661, A013663, A013667, A013669, A013671, A013675, A013677, A059956 (6/Pi^2), A084225; A084226.

Cf. A197070: 3*zeta(3)/4; A233090: 5*zeta(3)/8; A233091: 7*zeta(3)/8.

Cf. A001008, A002805, A143003, A143007.

Cf. A000578 (cubes).

Cf. sums of inverses: A152623 (tetrahedral numbers), A175577 (octahedral numbers), A295421 (dodecahedral numbers), A175578 (icosahedral numbers).

Sequence in context: A308214 A327371 A037228 * A042970 A158327 A136581

Adjacent sequences: A002114 A002115 A002116 * A002118 A002119 A002120

Keyword

cons,nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Additional comments from Robert G. Wilson v, Dec 08 2000

Quotation from Stan Wagon corrected by N. J. A. Sloane on Dec 24 2005. Thanks to Jose Brox for noticing this error.

Edited by M. F. Hasler, Sep 26 2017

Status

approved