1,2

Sometimes called Apery's constant.

"A natural question is whether Zeta(3) is a rational multiple of Pi^3. This is not known, though in 1978 R. Apery 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

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

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.

Hardy and Wright, 'An Introduction to the Theory of Numbers' pp. 47,268-269

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

Yaglom and Yaglom, 'Challenging Mathematical Problems with Elementary Solutions' ex. 92-93

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

T. Amdeberhan, Faster and Faster convergent series for zeta(3)

Author?, Probability of Random Numbers Being Coprime

Author?, Probability of two numbers being coprime

J. Borwein and D. Bradley, Empirically determined Ap'ery-like formulae for zeta(4n+3)

L. Euler, On the sums of series of reciprocals

L. Euler, De summis serierum reciprocarum, E41.

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

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

M. Kondratiewa and S. Sadov, Markov's transformation of series and the WZ method

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

S. Plouffe, Zeta(3) or Apery's constant to 2000 places

A. van der Poorten, A Proof that Euler Missed

Tanguy Rivoal, Title?

G. Villemin's Almanach of Numbers, Apery's Constant(Text 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, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, 'Relatively Prime'

Wikipedia, Riemann zeta function

H. Wilf, Accelerated series for universal constants, by the WZ method

Wadim Zudilin, An elementary proof of Apery's theorem

F. M. S. Lima, A simple approximate expression for the Ape'ry's constant accurate to 21 digits, Oct 14, 2009 [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009]

Lima conjectures that zeta(3) = (-5/197) + (11/394)*(pi^2)*(ln(2)) + (283/394)*(pi)*(ln(2)^2) + (236/197)*(ln(3)^3) + (209/394)*(ln(1+sqrt(2)^3) + (93*pi*gamma)/197 where gamma is the Euler-Mascheroni constant. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009]

1.2020569031595942853997...

RealDigits[ N[ Zeta[3], 100] ] [ [1] ]

(PARI) { default(realprecision, 20080); x=zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002117.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 19 2009]

Cf. A013631, A013679, A013661, A013663, A013667, A013669, A013671, A013675, A013677.

Cf. A059956 for 6/Pi^2.

Cf. A084225; A084226.

Adjacent sequences: A002114 A002115 A002116 this_sequence A002118 A002119 A002120

Sequence in context: A011420 A035686 A037228 this_sequence A042970 A158327 A136581

cons,nonn,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David W. Wilson (davidwwilson(AT)comcast.net). Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 08 2000

Quotation from Stan Wagon corrected by N. J. A. Sloane (njas(AT)research.att.com) on Dec 24 2005. Thanks to Jose Brox for noticing this error.

Fixed PARI Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 17 2009

New arXiv paper conjectures rational approximation for zeta(3). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 14 2009]