Binary Binary numeral system The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
Decimal Decimal The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
Hexadecimal Hexadecimal In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
Continued fraction Continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...	Note that this continuing fraction is not periodic.

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Apéry's constant is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in conjunction with the gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 when solving certain integrals involving exponential functions in a quotient which appear occasionally in physics, for instance when evaluating the two-dimensional case of the Debye model

Debye model

In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat in a solid. It treats the vibrations of the atomic lattice as phonons in a box, in contrast to the Einstein model, which treats the...

.
It is defined as the number ζ(3),


where ζ is the Riemann zeta function. It has an approximate value of

ζ(3) =   .

The reciprocal of this constant

Constant (mathematics)

In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...

 is the probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

 that any three positive integers, chosen at random, will be relatively prime (in the sense that as N goes to infinity, the probability that three positive integers less than N chosen uniformly at random will be relatively prime approaches this value).

Apéry's theorem

This value was named for Roger Apéry

Roger Apéry

Roger Apéry was a Greek-French mathematician most remembered for Apéry's theorem, that ζ is an irrational number where ζ denotes the Riemann zeta function....

 (1916–1994), who in 1978 proved it to be irrational

Irrational number

In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....

. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and shorter proofs have been found later, using Legendre polynomials. It is not known whether Apéry's constant is transcendental

Transcendental number

In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

.

Work by Wadim Zudilin

Wadim Zudilin

Wadim Zudilin is a Russian number theorist who is active in studying hypergeometric functions and zeta constants. He studied under Yuri V. Nesterenko and worked at Moscow State University, the Steklov Institute of Mathematics and the Max Planck Institute for Mathematics. He now works at the...

 and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2n+1) must be irrational, and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.

Series representation

In 1772, Leonhard Euler

Leonhard Euler

Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

  gave the series representation :


which was subsequently rediscovered several times.

Ramanujan gives several series, which are notable in that they can provide several digits of accuracy per iteration. These include:


Simon Plouffe

Simon Plouffe

Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...

 has developed other series:


Similar relations for the values of are given in the article zeta constants.

Many additional series representations have been found, including:







and


where


Some of these have been used to calculate Apéry's constant with several million digits.

gives a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time, and logarithmic space.

Other formulas

Apéry's constant can be expressed in terms of the second-order polygamma function as


It can be expressed as the non-periodic continued fraction

Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

 [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] .

Known digits

The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.

Number of known decimal digits of Apéry's constant ζ(3)

Date	Decimal digits	Computation performed by
1735	16	Leonhard Euler Leonhard Euler Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
unknown	16	Adrien-Marie Legendre Adrien-Marie Legendre Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
1887	32	Thomas Joannes Stieltjes Thomas Joannes Stieltjes Thomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France. He was a pioneer in the field of moment problems and contributed to the study of continued fractions....
1996	520,000	Greg J. Fee & Simon Plouffe Simon Plouffe Simon Plouffe is a Quebec mathematician born on June 11, 1956 in Saint-Jovite, Quebec. He discovered the formula for the BBP algorithm which permits the computation of the nth binary digit of π, in 1995...
1997	1,000,000	Bruno Haible & Thomas Papanikolaou
May 1997	10,536,006	Patrick Demichel
February 1998	14,000,074	Sebastian Wedeniwski
March 1998	32,000,213	Sebastian Wedeniwski
July 1998	64,000,091	Sebastian Wedeniwski
December 1998	128,000,026	Sebastian Wedeniwski
September 2001	200,001,000	Shigeru Kondo & Xavier Gourdon
February 2002	600,001,000	Shigeru Kondo & Xavier Gourdon
February 2003	1,000,000,000	Patrick Demichel & Xavier Gourdon
April 2006	10,000,000,000	Shigeru Kondo & Steve Pagliarulo (see )
January 2009	15,510,000,000	Alexander J. Yee & Raymond Chan (see )
March 2009	31,026,000,000	Alexander J. Yee & Raymond Chan (see )
September 2010	100,000,001,000	Alexander J. Yee (see Yee)