Catalan's constant

File:Catalan constant area.png

catalan constant as area under the curve of arctanx /x

Template:CS1 config In mathematics, Catalan's constant Template:Mvar, is the alternating sum of the reciprocals of the odd square numbers, being defined by:

<math>G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \frac{1}{9^2} - \cdots,</math>

where Template:Mvar is the Dirichlet beta function. Its numerical value<ref>Template:Cite book</ref> is approximately (sequence A006752 in the OEIS)

Template:Math

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.<ref>Template:Citation</ref><ref>Template:Citation</ref>

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.<ref>Template:Citation.</ref> It is 1/8 of the volume of the complement of the Borromean rings.<ref>Template:Citation</ref>

In combinatorics and statistical mechanics, it arises in connection with counting domino tilings,<ref>Template:Citation</ref> spanning trees,<ref>Template:Citation</ref> and Hamiltonian cycles of grid graphs.<ref>Template:Citation</ref>

In number theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form <math>n^2+1</math> 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.<ref>Template:Citation</ref>

Catalan's constant also appears in the calculation of the mass distribution of spiral galaxies.<ref>Template:Citation</ref><ref>Template:Citation</ref>

Properties

Template:Unsolved It is not known whether Template:Mvar is irrational, let alone transcendental.<ref>Template:Citation.</ref> Template:Mvar has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven".<ref>Template:Citation</ref>

There exist however partial results. It is known that infinitely many of the numbers β(2n) are irrational, where β(s) is the Dirichlet beta function.<ref>Template:Cite journal</ref> In particular at least one of β(2), β(4), β(6), β(8), β(10) and β(12) must be irrational, where β(2) is Catalan's constant.<ref>Template:Cite journal</ref> These results by Wadim Zudilin and Tanguy Rivoal are related to similar ones given for the odd zeta constants ζ(2n+1).

Catalan's constant is known to be an algebraic period, which follows from some of the double integrals given below.

Series representations

Catalan's constant appears in the evaluation of several rational series including:<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><math display="block">\frac{\pi^2}{16}+\frac G2 = \sum_{n=0}^\infty \frac{1}{(4n+1)^2}.</math><math display="block">\frac{\pi^2}{16}-\frac G2 = \sum_{n=0}^\infty \frac{1}{(4n+3)^2}.</math> The following two formulas involve quickly converging series, and are thus appropriate for numerical computation: <math display="block">\begin{align} G & = 3 \sum_{n=0}^\infty \frac{1}{2^{4n}} \left(-\frac{1}{2(8n+2)^2}+\frac{1}{2^2(8n+3)^2}-\frac{1}{2^3(8n+5)^2}+\frac{1}{2^3(8n+6)^2}-\frac{1}{2^4(8n+7)^2}+\frac{1}{2(8n+1)^2}\right) \\ & \qquad -2 \sum_{n=0}^\infty \frac{1}{2^{12n}} \left(\frac{1}{2^4(8n+2)^2}+\frac{1}{2^6(8n+3)^2}-\frac{1}{2^9(8n+5)^2}-\frac{1}{2^{10} (8n+6)^2}-\frac{1}{2^{12} (8n+7)^2}+\frac{1}{2^3(8n+1)^2}\right) \end{align}</math> and <math display="block">G = \frac{\pi}{8}\log\left(2 + \sqrt{3}\right) + \frac{3}{8}\sum_{n=0}^\infty \frac{1}{(2n+1)^2 \binom{2n}{n}}.</math>

The theoretical foundations for such series are given by Broadhurst, for the first formula,<ref>Template:Cite arXiv</ref> and Ramanujan, for the second formula.<ref>Template:Cite book</ref> The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.<ref>Template:Cite journal</ref><ref>Template:Cite book</ref> Using these series, calculating Catalan's constant is now about as fast as calculating Apéry's constant, <math>\zeta(3)</math>.<ref name="Yee_formulas" />

Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:<ref name="Yee_formulas">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>G = \frac{1}{2}\sum_{k=0}^{\infty }\frac{(-8)^{k}(3k+2)}{(2k+1)^{3}{\binom{2k}{k}}^{3}}</math>

<math>G = \frac{1}{64}\sum_{k=1}^{\infty }\frac{256^{k}(580k^2-184k+15)}{k^3(2k-1)\binom{6k}{3k}\binom{6k}{4k}\binom{4k}{2k}}</math>

<math>G = -\frac{1}{1024}\sum_{k=1}^{\infty }\frac{(-4096)^k(45136k^4-57184k^3+21240k^2-3160k+165)}{k^3(2k-1)^3}\left( \frac{(2k)!^6(3k)!^3}{k!^3(6k)!^3} \right)</math>

All of these series have time complexity <math>O(n\log(n)^3)</math>.<ref name="Yee_formulas"/>

Integral identities

As Seán Stewart writes, "There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant."<ref>Template:Citation</ref> Some of these expressions include: <math display="block">\begin{align} G &= -\frac{1}{\pi i}\int_{0}^{\frac{\pi}{2}} \ln\ln \tan x \ln \tan x \,dx \\[3pt] G &= \iint_{[0,1]^2} \! \frac{1}{1+x^2 y^2} \,dx\, dy \\[3pt] G &= \int_0^1\int_0^{1-x} \frac{1}{1 -x^2-y^2} \,dy\,dx \\[3pt] G &= \int_1^\infty \frac{\ln t}{1 + t^2} \,dt \\[3pt] G &= -\int_0^1 \frac{\ln t}{1 + t^2} \,dt \\[3pt] G &= \frac{1}{2} \int_0^\frac{\pi}{2} \frac{t}{\sin t} \,dt \\[3pt] G &= \int_0^\frac{\pi}{4} \ln \cot t \,dt \\[3pt] G &= \frac{1}{2} \int_0^\frac{\pi}{2} \ln \left( \sec t +\tan t \right) \,dt \\[3pt] G &= \int_0^1 \frac{\arccos t}{\sqrt{1+t^2}} \,dt \\[3pt] G &= \int_0^1 \frac{\operatorname{arcsinh} t}{\sqrt{1-t^2}} \,dt \\[3pt] G &= \frac{1}{2} \int_0^\infty \frac{\operatorname{arctan} t}{t\sqrt{1+t^2}} \,dt \\[3pt] G &= \frac{1}{2} \int_0^1 \frac{\operatorname{arctanh} t}{\sqrt{1-t^2}} \,dt \\[3pt] G &= \int_0^\infty \arccot e^{t} \,dt \\[3pt] G &= \frac{1}{4} \int_0^{{\pi^2}/{4}} \csc \sqrt{t} \,dt \\[3pt] G &= \frac{1}{16} \left(\pi^2 + 4\int_1^\infty \arccsc^2 t \,dt\right) \\[3pt] G &= \frac{1}{2} \int_0^\infty \frac{t}{\cosh t} \,dt \\[3pt] G &= \frac{\pi}{2} \int_1^\infty \frac{\left(t^4-6t^2+1\right)\ln\ln t}{\left(1+t^2\right)^3} \,dt \\[3pt] G &= \frac{1}{2} \int_0^\infty \frac{\arcsin \left(\sin t\right)}{t} \,dt \\[3pt] G &= 1 + \lim_{\alpha\to{1^-}}\!\left\{\int_0^{\alpha}\!\frac{\left(1+6t^2+t^4\right)\arctan{t}}{t\left(1-t^2\right)^2}\, dt + 2\operatorname{artanh}{\alpha} - \frac{\pi\alpha}{1-\alpha^2} \right\} \\[3pt] G &= 1 - \frac18 \iint_{\R^2}\!\!\frac{x\sin\left(2xy/\pi\right)}{\,\left(x^2+\pi^2\right)\cosh x\sinh y\,} \,dx\,dy \\[3pt] G &= \int_{0}^{\infty}\int_{0}^{\infty}\frac{\sqrt[4]{x} \left(\sqrt{x} \sqrt{y}-1\right)}{(x+1)^2 \sqrt[4]{y} (y+1)^2 \log (x y)}dxdy \end{align}</math>

where the last three formulas are related to Malmsten's integrals.<ref>Template:Cite journal</ref>

If Template:Math is the complete elliptic integral of the first kind, as a function of the elliptic modulus Template:Math, then <math display="block"> G = \tfrac{1}{2} \int_0^1 \mathrm{K}(k)\,dk </math>

If Template:Math is the complete elliptic integral of the second kind, as a function of the elliptic modulus Template:Math, then <math display="block"> G = -\tfrac{1}{2}+\int_0^1 \mathrm{E}(k)\,dk </math>

With the gamma function Template:Math <math display="block">\begin{align} G &= \frac{\pi}{4} \int_0^1 \Gamma\left(1+\frac{x}{2}\right)\Gamma\left(1-\frac{x}{2}\right)\,dx \\ &= \frac{\pi}{2} \int_0^\frac12\Gamma(1+y)\Gamma(1-y)\,dy \end{align}</math>

The integral <math display="block"> G = \operatorname{Ti}_2(1)=\int_0^1 \frac{\arctan t}{t}\,dt </math> is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.

Relation to special functions

Template:Mvar appears in values of the second polygamma function, also called the trigamma function, at fractional arguments:<ref name=":0" />

<math display="block">\begin{align} \psi_1 \left(\tfrac14\right) &= \pi^2 + 8G \\ \psi_1 \left(\tfrac34\right) &= \pi^2 - 8G. \end{align}</math>

Simon Plouffe gives an infinite collection of identities between the trigamma function, Template:Pi² 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 G-function|Barnes Template:Mvar-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 Template:Mvar-function, the following expression is obtained (see Clausen function for more):

<math display="block">G=4\pi \log\left( \frac{ G\left(\frac{3}{8}\right) G\left(\frac{7}{8}\right) }{ G\left(\frac{1}{8}\right) G\left(\frac{5}{8}\right) } \right) +4 \pi \log \left( \frac{ \Gamma\left(\frac{3}{8}\right) }{ \Gamma\left(\frac{1}{8}\right) } \right) +\frac{\pi}{2} \log \left( \frac{1+\sqrt{2} }{2 \left(2-\sqrt{2}\right)} \right).</math>

If one defines the Lerch transcendent Template:Math by <math display="block">\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s},</math> then <math display="block"> G = \tfrac{1}{4}\Phi\left(-1, 2, \tfrac{1}{2}\right).</math>

Continued fraction

Template:Mvar can be expressed in the following form:<ref>Template:Cite journal</ref>

<math>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}}}}}}</math>

The simple continued fraction is given by:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>G=\cfrac{1}{1+\cfrac{1}{10+\cfrac{1}{1+\cfrac{1}{8+\cfrac{1}{1+\cfrac{1}{88+\ddots}}}}}}</math>

This continued fraction would have infinite terms if and only if <math>G</math> is irrational, which is still unresolved.

Known digits

The number of known digits of Catalan's constant Template:Mvar has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.<ref name=Gourdon>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Number of known decimal digits of Catalan's constant Template:Mvar

Date	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	Template:Val	Greg J. Fee
1996	Template:Val	Greg J. Fee
August 14, 1996	Template:Val	Greg J. Fee & Simon Plouffe
September 29, 1996	Template:Val	Thomas Papanikolaou
1996	Template:Val	Thomas Papanikolaou
1997	Template:Val	Patrick Demichel
January 4, 1998	Template:Val	Xavier Gourdon
2001	Template:Val	Xavier Gourdon & Pascal Sebah
2002	Template:Val	Xavier Gourdon & Pascal Sebah
October 2006	Template:Val	citation	CitationClass=web }}</ref>
August 2008	Template:Val	Shigeru Kondo & Steve Pagliarulo<ref name=Gourdon />
January 31, 2009	Template:Val	citation	CitationClass=web }}</ref>
April 16, 2009	Template:Val	Alexander J. Yee & Raymond Chan<ref name=yee_chan/>
June 7, 2015	Template:Val	citation	CitationClass=web }}</ref>
April 12, 2016	Template:Val	Ron Watkins<ref name=setti/>
February 16, 2019	Template:Val	Tizian Hanselmann<ref name=setti/>
March 29, 2019	Template:Val	Mike A & Ian Cutress<ref name=setti/>
July 16, 2019	Template:Val	citation	CitationClass=web }}</ref><ref name=kim>{{#invoke:citation/CS1|citation	CitationClass=web }}</ref>
September 6, 2020	Template:Val	citation	CitationClass=web }}</ref>
March 9, 2022	Template:Val	Seungmin Kim<ref name=yrecord/>

See also

• Gieseking manifold
• List of mathematical constants
• Mathematical constant
• Particular values of Riemann zeta function

Further reading

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External links

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• Template:Cite book (Provides the first 300,000 digits of Catalan's constant)
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