# Apéry's constant

In mathematics, Apéry's constant is the infinite sum of the reciprocals of the positive integers, cubed. That is, it is defined as the number
    ${\begin{aligned}\zeta (3)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}\\&=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right),\end{aligned}}$
where ζ is the Riemann zeta function. It has an approximate value of
    ζ(3) ≈ 1.202056903159594285399738161511449990764986292… (sequence A002117 in the OEIS).
It is named after Roger Apéry, who proved that it is an irrational number.

## Uses

Apéry's constant 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 the analysis of random minimum spanning trees and in conjunction with the gamma function 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 and the Stefan–Boltzmann law.
The reciprocal of ζ(3) (0.8319073725807... (sequence A088453 in the OEIS)) is the probability that any three positive integers, chosen at random, will be relatively prime, in the sense that as N approaches infinity, the probability that three positive integers less than N chosen uniformly at random will not share a common prime factor approaches this value. (The probability for n positive integers is 1/ζ(n).) In the same sense, it is the probability that a positive integer chosen at random will not be evenly divisible by the cube of an integer greater than one. (The probability for not having divisibility by an n-th power is 1/ζ(n).)

## Properties

ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later.
Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for ζ(3),
    $\zeta (3)=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\frac {1}{1-xyz}}\,dx\,dy\,dz,$
by the Legendre polynomials. In particular, van der Poorten's article chronicles this approach by noting that
    $I_{3}:=-{\frac {1}{2}}\int _{0}^{1}\int _{0}^{1}{\frac {P_{n}(x)P_{n}(y)\log(xy)}{1-xy}}\,dx\,dy=b_{n}\zeta (3)-a_{n},$
where $|I|\leq \zeta (3)(1-{\sqrt {2}})^{4n}$, $P_{n}(z)$ are the Legendre polynomials, and the subsequences $b_{n},2\operatorname {lcm} (1,2,\ldots ,n)\cdot a_{n}\in \mathbb {Z}$ are integers or almost integers.
Many people have tried to extend Apéry's proof that ζ(3) is irrational to other values of the Riemann zeta function with odd arguments. Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants ζ(2n + 1) are irrational. In particular at least one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.
Apéry's constant has not yet been proved transcendental, but it is known to be an algebraic period. This follows immediately from the form of its triple integral.

## Series representations
### Classical

In addition to the fundamental series:
    $\zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{k^{3}}},$
Leonhard Euler gave the series representation:
    $\zeta (3)={\frac {\pi ^{2}}{7}}\left(1-4\sum _{k=1}^{\infty }{\frac {\zeta (2k)}{2^{2k}(2k+1)(2k+2)}}\right)$
in 1772, which was subsequently rediscovered several times.

### Fast convergence

Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of ζ(3). Since the 1990s, this search has focused on computationally efficient series with fast convergence rates (see section "Known digits").
The following series representation was found by A. A. Markov in 1890, rediscovered by Hjortnaes in 1953, and rediscovered once more and widely advertised by Apéry in 1979:
    $\zeta (3)={\frac {5}{2}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {k!^{2}}{(2k)!k^{3}}}.$
The following series representation gives (asymptotically) 1.43 new correct decimal places per term:
    $\zeta (3)={\frac {1}{4}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {(k-1)!^{3}(56k^{2}-32k+5)}{(2k-1)^{2}(3k)!}}.$
The following series representation gives (asymptotically) 3.01 new correct decimal places per term:
    $\zeta (3)={\frac {1}{64}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {k!^{10}(205k^{2}+250k+77)}{(2k+1)!^{5}}}.$
The following series representation gives (asymptotically) 5.04 new correct decimal places per term:
    $\zeta (3)={\frac {1}{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}.$
It has been used to calculate Apéry's constant with several million correct decimal places.
The following series representation gives (asymptotically) 3.92 new correct decimal places per term:
    $\zeta (3)={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}.$

### Digit by digit

In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained by a spigot algorithm in nearly linear time and logarithmic space.

### Thue-Morse sequence

Apéry's constant can be represented in terms of the Thue-Morse sequence $(t_{n})_{n\geq 0}$, as follows:
    ${\begin{aligned}\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}$
This is a special case of the following formula (valid for all $s$ with real part greater than $1$):
    $(2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).$

### Others

The following series representation was found by Ramanujan:
    $\zeta (3)={\frac {7}{180}}\pi ^{3}-2\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}.$
The following series representation was found by Simon Plouffe in 1998:
    $\zeta (3)=14\sum _{k=1}^{\infty }{\frac {1}{k^{3}\sinh(\pi k)}}-{\frac {11}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}-{\frac {7}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}+1)}}.$
Srivastava (2000) collected many series that converge to Apéry's constant.

## Integral representations

There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.

### Simple formulas

The following formula follows directly from the integral definition of the zeta function:
    $\zeta (3)={\frac {1}{2}}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}\,dx$

### More complicated formulas

Other formulas include
    $\zeta (3)=\pi \int _{0}^{\infty }{\frac {\cos(2\arctan x)}{(x^{2}+1)\left(\cosh {\frac {1}{2}}\pi x\right)^{2}}}\,dx$
and
    $\zeta (3)=-{\frac {1}{2}}\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(xy)}{1-xy}}\,dx\,dy=-\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(1-xy)}{xy}}\,dx\,dy.$
Also,
    ${\begin{aligned}\zeta (3)&={\frac {8\pi ^{2}}{7}}\int _{0}^{1}{\frac {x(x^{4}-4x^{2}+1)\log \log {\frac {1}{x}}}{(1+x^{2})^{4}}}\,dx\\&={\frac {8\pi ^{2}}{7}}\int _{1}^{\infty }{\frac {x(x^{4}-4x^{2}+1)\log \log {x}}{(1+x^{2})^{4}}}\,dx.\end{aligned}}$
A connection to the derivatives of the gamma function
    $\zeta (3)=-{\tfrac {1}{2}}(\Gamma '''(1)+\gamma ^{3}+{\tfrac {1}{2}}\pi ^{2}\gamma )=-{\tfrac {1}{2}}\psi ^{(2)}(1)$
is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma functions.

## Continued fraction

Apéry's constant is related to the following continued fraction:
    ${\frac {6}{\zeta (3)}}=5-{\cfrac {1}{117-{\cfrac {64}{535-{\cfrac {729}{1436-{\cfrac {4096}{3105-{\cfrac {15625}{\dots }}}}}}}}}}$
with $a_{n}=34n^{3}+51n^{2}+27n+5$ and $b_{n}=-n^{6}$.
Its simple continued fraction is given by:
    $\zeta (3)=1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{18+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\dots }}}}}}}}}}}}$

## Known digits

The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades, and now stands at more than 2×10$^{12}$. This is due both to the increasing performance of computers and to algorithmic improvements.
    Number of known decimal digits of Apéry's constant ζ(3) DateDecimal digitsComputation performed by 173516Leonhard Euler Unknown16Adrien-Marie Legendre 188732Thomas Joannes Stieltjes 1996520000Greg J. Fee & Simon Plouffe 19971000000Bruno Haible & Thomas Papanikolaou May 199710536006Patrick Demichel February 199814000074Sebastian Wedeniwski March 199832000213Sebastian Wedeniwski July 199864000091Sebastian Wedeniwski December 1998128000026Sebastian Wedeniwski September 2001200001000Shigeru Kondo & Xavier Gourdon February 2002600001000Shigeru Kondo & Xavier Gourdon February 20031000000000Patrick Demichel & Xavier Gourdon April 200610000000000Shigeru Kondo & Steve Pagliarulo January 21, 200915510000000Alexander J. Yee & Raymond Chan February 15, 200931026000000Alexander J. Yee & Raymond Chan September 17, 2010100000001000Alexander J. Yee September 23, 2013200000001000Robert J. Setti August 7, 2015250000000000Ron Watkins December 21, 2015400000000000Dipanjan Nag August 13, 2017500000000000Ron Watkins May 26, 20191000000000000Ian Cutress July 26, 20201200000000100Seungmin Kim December 22, 20232020569031595Andrew Sun