Catalan's constant explained

In mathematics, Catalan's constant, is defined by

G=\beta(2)=

	infty
\sum
	n=0

	(-1)ⁿ
	(2n+1)²

	1
	1²

	1
	3²

	1
	5²

	1
	7²

	1
	9²

- … ,

where is the Dirichlet beta function. Its numerical value^[1] is approximately

It is not known whether is irrational, let alone transcendental.^[2] has been called "arguably the most basic constant whose irrationality and transcendence (though stronglysuspected) remain unproven".

Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865.

Uses

In low-dimensional topology, Catalan's constant is 1/4 of the volume of an ideal hyperbolic octahedron, and therefore 1/4 of the hyperbolic volume of the complement of the Whitehead link.^[3] It is 1/8 of the volume of the complement of the Borromean rings.

In combinatorics and statistical mechanics, it arises in connection with counting domino tilings, spanning trees, and Hamiltonian cycles of grid graphs.

In number theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form

n²⁺¹

according to Hardy and Littlewood's Conjecture F. However, it is an unsolved problem (one of Landau's problems) whether there are even infinitely many primes of this form.

Catalan's constant also appears in the calculation of the mass distribution of spiral galaxies.

Known digits

The number of known digits of Catalan's constant has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.^[4]


Decimal digits	Computation performed by
1832	16	Thomas Clausen
1858	19	Carl Johan Danielsson Hill
1864	14	Eugène Charles Catalan
1877	20	James W. L. Glaisher
1913	32	James W. L. Glaisher
1990		Greg J. Fee
1996		Greg J. Fee
August 14, 1996		Greg J. Fee & Simon Plouffe
September 29, 1996		Thomas Papanikolaou
1996		Thomas Papanikolaou
1997		Patrick Demichel
January 4, 1998		Xavier Gourdon
2001		Xavier Gourdon & Pascal Sebah
2002		Xavier Gourdon & Pascal Sebah
October 2006		Shigeru Kondo & Steve Pagliarulo^[5]
August 2008		Shigeru Kondo & Steve Pagliarulo
January 31, 2009		Alexander J. Yee & Raymond Chan^[6]
April 16, 2009		Alexander J. Yee & Raymond Chan
June 7, 2015		Robert J. Setti^[7]
April 12, 2016		Ron Watkins
February 16, 2019		Tizian Hanselmann
March 29, 2019		Mike A & Ian Cutress
July 16, 2019		Seungmin Kim^[8] ^[9]
September 6, 2020		Andrew Sun^[10]
March 9, 2022		Seungmin Kim

Integral identities

As Seán Stewart writes, "There is a rich and seemingly endless source of definite integrals thatcan be equated to or expressed in terms of Catalan's constant." Some of these expressions include: $\beginG &= -\frac\int_^ \ln\ln \tan x \ln \tan x \,dx \\[3pt]G &= \iint_ \! \frac \,dx\, dy \\[3pt] G &= \int_0^1\int_0^ \frac \,dy\,dx \\[3pt]G &= \int_1^\infty \frac \,dt \\[3pt]G &= -\int_0^1 \frac \,dt \\[3pt]G &= \frac \int_0^\frac \frac \,dt \\[3pt]G &= \int_0^\frac \ln \cot t \,dt \\[3pt]G &= \frac \int_0^\frac \ln \left(\sec t +\tan t \right) \,dt \\[3pt]G &= \int_0^1 \frac \,dt \\[3pt]G &= \int_0^1 \frac \,dt \\[3pt]G &= \frac \int_0^\infty \frac \,dt \\[3pt]G &= \frac \int_0^1 \frac \,dt \\[3pt]G &= \int_0^\infty \arccot e^ \,dt \\[3pt]G &= \frac \int_0^ \csc \sqrt \,dt \\[3pt]G &= \frac \left(\pi^2 + 4\int_1^\infty \arccsc^2 t \,dt\right) \\[3pt]G &= \frac \int_0^\infty \frac \,dt \\[3pt]G &= \frac \int_1^\infty \frac \,dt \\[3pt]G &= \frac \int_0^\infty \frac \,dt \\[3pt]G &= 1 + \lim_\!\left\ \\[3pt]G &= 1 - \frac18 \iint_\!\!\frac \,dx\,dy \\[3pt]G &= \int_^\int_^\fracdxdy\end$

where the last three formulas are related to Malmsten's integrals.^[11]

If is the complete elliptic integral of the first kind, as a function of the elliptic modulus, then $G = \tfrac \int_0^1 \mathrm(k)\,dk$

If is the complete elliptic integral of the second kind, as a function of the elliptic modulus, then $G = -\tfrac+\int_0^1 \mathrm(k)\,dk$

With the gamma function $\beginG &= \frac \int_0^1 \Gamma\left(1+\frac\right)\Gamma\left(1-\frac\right)\,dx \\&= \frac \int_0^\frac12\Gamma(1+y)\Gamma(1-y)\,dy\end$

The integral $G = \operatorname_2(1)=\int_0^1 \frac\,dt$ is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.

Relation to other special functions

appears in values of the second polygamma function, also called the trigamma function, at fractional arguments:

$\begin\psi_1 \left(\tfrac14\right) &= \pi^2 + 8G \\\psi_1 \left(\tfrac34\right) &= \pi^2 - 8G.\end$

Simon Plouffe gives an infinite collection of identities between the trigamma function, ² and Catalan's constant; these are expressible as paths on a graph.

Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes -function, as well as integrals and series summable in terms of the aforementioned functions.

As a particular example, by first expressing the inverse tangent integral in its closed form – in terms of Clausen functions – and then expressing those Clausen functions in terms of the Barnes -function, the following expression is obtained (see Clausen function for more):

$G=4\pi \log\left(\frac \right) +4 \pi \log \left(\frac \right) +\frac \log \left(\frac \right).$

If one defines the Lerch transcendent (related to the Lerch zeta function) by $\Phi(z, s, \alpha) = \sum_^\infty \frac,$ then $G = \tfrac\Phi\left(-1, 2, \tfrac\right).$

Quickly converging series

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation: $\beginG & = 3 \sum_^\infty \frac\left(-\frac+\frac-\frac+\frac-\frac+\frac\right)- \\& \qquad -2 \sum_^\infty \frac\left(\frac+\frac-\frac-\frac-\frac+\frac\right)\end$ and $G = \frac\log\left(2 + \sqrt\right) + \frac\sum_^\infty \frac.$

The theoretical foundations for such series are given by Broadhurst, for the first formula,^[12] and Ramanujan, for the second formula.^[13] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.^[14] ^[15] Using these series, calculating Catalan's constant is now about as fast as calculating Apery's constant,

\zeta(3)

Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:^[16]

	1
	2

	infty
\sum
	k=0

	(-8)^k(3k+2)
	(2k+1)³{\binom{2k

{k}}³

}

	1
	64

	infty
\sum
	k=1

	256^k(580k^2-184k+15)
	k^{3(2k-1)\binom{6k}

{3k}\binom{6k}{4k}\binom{4k}{2k}}

G=-

	1
	1024

	infty
\sum
	k=1

	(-4096)^k(45136k^4-57184k^3+21240k^2-3160k+165)
	k^3(2k-1)³

\left(

	(2k)!^6(3k)!³
	k!^3(6k)!³

\right)

O(nlog(n)³⁾

.^[16]

Continued fraction

can be expressed in the following form^[17]

G=\cfrac{1}{1+\cfrac{1^{4}{8+\cfrac{3}^{4}{16+\cfrac{5}^{4}{24+\cfrac{7}^{4}{32+\cfrac{9}^{4}{40+\ddots}}}}}}}

The simple continued fraction is given by^[18]

G=\cfrac{1}{1+\cfrac{1}{10+\cfrac{1}{1+\cfrac{1}{8+\cfrac{1}{1+\cfrac{1}{88+\ddots}}}}}}

This continued fraction would have infinite terms if and only if

is irrational, which is still unresolved.

External links

Web site: Victor . Adamchik . 33 representations for Catalan's constant . https://web.archive.org/web/20160807111945/https://www.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm . 2016-08-07 . 14 July 2005 .
Web site: Simon . Plouffe . A few identities (III) with Catalan . 1993 . https://web.archive.org/web/20190626124124/https://lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html . 2019-06-26 . 29 July 2005 . (Provides over one hundred different identities).
Web site: Simon . Plouffe . A few identities with Catalan constant and Pi^2 . 1999 . https://web.archive.org/web/20190626124128/https://lacim.uqam.ca/~plouffe/IntegerRelations/identities3.html . 2019-06-26 . 29 July 2005 . (Provides a graphical interpretation of the relations)
Book: Fee , Greg . Catalan's Constant (Ramanujan's Formula) . 1996 . (Provides the first 300,000 digits of Catalan's constant)
- Web site: Fredrik . Johansson . 0.915965594177219015054603514932 . Ordner, a catalog of real numbers in Fungrim . 21 April 2021 .
Web site: Catalan's Constant . Let's Learn, Nemo! . 10 August 2020 . . 6 April 2021 .

Notes and References

Book: Papanikolaou. Thomas. Catalan's Constant to 1,500,000 Places. Gutenberg.org. March 1997.
.
.
Web site: Gourdon. X.. Sebah. P.. Constants and Records of Computation. 11 September 2007.
Web site: Shigeru Kondo's website . 2008-01-31 . https://web.archive.org/web/20080211185703/http://ja0hxv.calico.jp/pai/ecatalan.html . 2008-02-11 . dead .
Web site: Large Computations . 31 January 2009.
Web site: Catalan's constant records using YMP . 14 May 2016.
Web site: Catalan's constant records using YMP . https://web.archive.org/web/20190722034426/http://www.numberworld.org/y-cruncher/ . 22 July 2019 . dead . 22 July 2019.
Web site: Catalan's constant world record by Seungmin Kim. 23 July 2019 . 17 October 2020.
Web site: Records set by y-cruncher. 2022-02-13. www.numberworld.org.
Iaroslav. Blagouchine. Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results. 2014. 10.1007/s11139-013-9528-5. 35. The Ramanujan Journal. 21–110. 120943474. 2018-10-01. https://web.archive.org/web/20181002020243/https://iblagouchine.perso.centrale-marseille.fr/publications/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).pdf. 2018-10-02. dead .
D. J. . Broadhurst. math.CA/9803067 . Polylogarithmic ladders, hypergeometric series and the ten millionth digits of and . 1998.
Book: Berndt, B. C.. Ramanujan's Notebook, Part I. Springer Verlag. 1985. 289. 978-1-4612-1088-7.
E. A.. Karatsuba. Fast evaluation of transcendental functions. Probl. Inf. Transm.. 27. 4. 339–360. 1991. 0754.65021. 1156939.
Book: Karatsuba, E. A.. Fast computation of some special integrals of mathematical physics. Scientific Computing, Validated Numerics, Interval Methods. limited. W.. Krämer. J. W.. von Gudenberg. 29–41. 2001. 10.1007/978-1-4757-6484-0_3.
Web site: Formulas and Algorithms. Alexander Yee. 14 May 2019. 5 December 2021.
Acta Arithmetica . 103 . 4 . 329–342 . Bowman, D. . Mc Laughlin, J.. amp . Polynomial continued fractions . English . 2002 . 10.4064/aa103-4-3 . 1812.08251 . 2002AcAri.103..329B . 119137246 . https://web.archive.org/web/20200413012537/https://www.wcupa.edu/sciences-mathematics/mathematics/jMcLaughlin/documents/4paper1.pdf . 2020-04-13 . live.
Web site: A014538 - OEIS . 2022-10-27 . oeis.org.