# Apéry's constant

Rationality Irrational ζ(3) 1.2020569031595942854...

In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number

{\displaystyle {\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[1]

ζ(3) ≈ 1.202056903159594285399738161511449990764986292 (sequence A002117 in the OEIS).

The constant is named after Roger Apéry. 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 the analysis of random minimum spanning trees[2] 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.

## Irrational number

Unsolved problem in mathematics:

Is Apéry's constant transcendental?

ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.[3] This result is known as Apéry's theorem. The original proof is complex and hard to grasp,[4] and simpler proofs were found later.[5]

Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for ζ(3),

${\displaystyle \zeta (3)=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\frac {1}{1-xyz}}\,dx\,dy\,dz,}$

by the Legendre polynomials. In particular, van der Poorten's article chronicles this approach by noting that

${\displaystyle I_{3}:=-{\frac {1}{2}}\int _{0}^{1}\int _{0}^{1}{\frac {P_{n}(x)P_{n}(y)\log(xy)}{1-xy}}\,dx\,dy=b_{n}\zeta (3)-a_{n},}$

where ${\displaystyle |I|\leq \zeta (3)(1-{\sqrt {2}})^{4n}}$, ${\displaystyle P_{n}(z)}$ are the Legendre polynomials, and the subsequences ${\displaystyle b_{n},2\operatorname {lcm} (1,2,\ldots ,n)\cdot a_{n}\in \mathbb {Z} }$ are integers or almost integers.

It is still not known whether Apéry's constant is transcendental.

## Series representations

### Classical

In addition to the fundamental series:

${\displaystyle \zeta (3)=\sum _{k=1}^{\infty }{\frac {1}{k^{3}}},}$

Leonhard Euler gave the series representation:[6]

${\displaystyle \zeta (3)={\frac {\pi ^{2}}{7}}\left(1-4\sum _{k=1}^{\infty }{\frac {\zeta (2k)}{2^{2k}(2k+1)(2k+2)}}\right)}$

in 1772, which was subsequently rediscovered several times.[7]

### 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,[8] rediscovered by Hjortnaes in 1953,[9] and rediscovered once more and widely advertised by Apéry in 1979:[3]

${\displaystyle \zeta (3)={\frac {5}{2}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {k!^{2}}{(2k)!k^{3}}}.}$

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:[10]

${\displaystyle \zeta (3)={\frac {1}{4}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {(k-1)!^{3}(56k^{2}-32k+5)}{(2k-1)^{2}(3k)!}}.}$

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:[11]

${\displaystyle \zeta (3)={\frac {1}{64}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {k!^{10}(205k^{2}+250k+77)}{(2k+1)!^{5}}}.}$

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:[12]

${\displaystyle \zeta (3)={\frac {1}{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}.}$

It has been used to calculate Apéry's constant with several million correct decimal places.[13]

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:[14]

${\displaystyle \zeta (3)={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}.}$

### 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 in nearly linear time and logarithmic space.[15]

### Thue-Morse sequence

The following representation was found by Tóth in 2022:[16]

{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}

where ${\displaystyle (t_{n})_{n\geq 0}}$ is the ${\displaystyle n^{\rm {th}}}$ term of the Thue-Morse sequence. In fact, this is a special case of the following formula (valid for all ${\displaystyle s}$ with real part greater than ${\displaystyle 1}$):

${\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).}$

### Others

The following series representation was found by Ramanujan:[17]

${\displaystyle \zeta (3)={\frac {7}{180}}\pi ^{3}-2\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}.}$

The following series representation was found by Simon Plouffe in 1998:[18]

${\displaystyle \zeta (3)=14\sum _{k=1}^{\infty }{\frac {1}{k^{3}\sinh(\pi k)}}-{\frac {11}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}-{\frac {7}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}+1)}}.}$

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:

${\displaystyle \zeta (3)={\frac {1}{2}}\int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}\,dx}$

### More complicated formulas

Other formulas include[19]

${\displaystyle \zeta (3)=\pi \int _{0}^{\infty }{\frac {\cos(2\arctan x)}{(x^{2}+1)\left(\cosh {\frac {1}{2}}\pi x\right)^{2}}}\,dx}$

and[20]

${\displaystyle \zeta (3)=-{\frac {1}{2}}\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(xy)}{1-xy}}\,dx\,dy=-\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(1-xy)}{xy}}\,dx\,dy.}$

Also,[21]

{\displaystyle {\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[22]

${\displaystyle \zeta (3)=-{\tfrac {1}{2}}(\Gamma '''(1)+\gamma ^{3}+{\tfrac {1}{2}}\pi ^{2}\gamma )=-{\tfrac {1}{2}}\psi ^{(2)}(1)}$

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma functions.[23]

## 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 2×1012. 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)
Date Decimal digits Computation performed by
1735 16 Leonhard Euler
1887 32 Thomas Joannes Stieltjes
1996 520000 Greg J. Fee & Simon Plouffe
1997 1000000 Bruno Haible & Thomas Papanikolaou
May 1997 10536006 Patrick Demichel
February 1998 14000074 Sebastian Wedeniwski
March 1998 32000213 Sebastian Wedeniwski
July 1998 64000091 Sebastian Wedeniwski
December 1998 128000026 Sebastian Wedeniwski[1]
September 2001 200001000 Shigeru Kondo & Xavier Gourdon
February 2002 600001000 Shigeru Kondo & Xavier Gourdon
February 2003 1000000000 Patrick Demichel & Xavier Gourdon[24]
April 2006 10000000000 Shigeru Kondo & Steve Pagliarulo
January 21, 2009 15510000000 Alexander J. Yee & Raymond Chan[25]
February 15, 2009 31026000000 Alexander J. Yee & Raymond Chan[25]
September 17, 2010 100000001000 Alexander J. Yee[26]
September 23, 2013 200000001000 Robert J. Setti[26]
August 7, 2015 250000000000 Ron Watkins[26]
December 21, 2015 400000000000 Dipanjan Nag[27]
August 13, 2017 500000000000 Ron Watkins[26]
May 26, 2019 1000000000000 Ian Cutress[28]
July 26, 2020 1200000000100 Seungmin Kim[28][29]
December 22, 2023 2020569031595 Andrew Sun[28]

## Reciprocal

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).[30]) 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).[30])

## Extension to ζ(2n + 1)

Many people have tried to extend Apéry's proof that ζ(3) is irrational to other values of the zeta function with odd arguments. Infinitely many of the numbers ζ(2n + 1) must be irrational,[31] and at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.[32]

## Notes

1. ^ a b
2. ^
3. ^ a b
4. ^
5. ^
6. ^
7. ^ Srivastava (2000), p. 571 (1.11).
8. ^
9. ^
10. ^
11. ^
12. ^ Wedeniwski (1998); Wedeniwski (2001). In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from Amdeberhan & Zeilberger (1997). The discovery year (1998) is mentioned in Simon Plouffe's Table of Records (8 April 2001).
13. ^
14. ^
15. ^
16. ^ Tóth, László (2022), "Linear Combinations of Dirichlet Series Associated with the Thue-Morse Sequence" (PDF), Integers, 22 (article 98), arXiv:2211.13570
17. ^ Berndt (1989, chapter 14, formulas 25.1 and 25.3).
18. ^
19. ^
20. ^
21. ^
22. ^ https://scipp.ucsc.edu/~haber/archives/physics116A10/psifun_10.pdf
23. ^ Evgrafov et al. (1969), exercise 30.10.1.
24. ^
25. ^ a b
26. ^ a b c d
27. ^
28. ^ a b c Yee, Alexander, Records set by y-cruncher, retrieved April 1, 2024.
29. ^ Apéry's constant world record by Seungmin Kim, 28 July 2020, retrieved July 28, 2020.
30. ^ a b
31. ^
32. ^

## References

• Ramaswami, V. (1934), "Notes on Riemann's ${\displaystyle \zeta }$-function", J. London Math. Soc., 9 (3): 165–169, doi:10.1112/jlms/s1-9.3.165.