Catalan's constant

File:Catalan constant area.png

Catalan constant as area under the curve of arctanx /x

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

$G=\beta (2)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1{)}^2}=\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\frac{1}{9^2}-\cdots ,$

where Template:Mvar is the Dirichlet beta function. Its numerical value 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.

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. 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 $n^2+1$ 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.

Properties

Unsolved problem in mathematics

Is Catalan's constant irrational? If so, is it transcendental?

More unsolved problems in mathematics

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

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. 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. 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:$$\frac{{\pi }^2}{16}+\frac{G}{2}=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(4n+1{)}^2}.$$$$\frac{{\pi }^2}{16}-\frac{G}{2}=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(4n+3{)}^2}.$$ The following two formulas involve quickly converging series, and are thus appropriate for numerical computation: $$\begin{array}{rl}G& =3\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^{4n}}(-\frac{1}{2(8n+2{)}^2}+\frac{1}{2^2(8n+3{)}^2}-\frac{1}{2^3(8n+5{)}^2}+\frac{1}{2^3(8n+6{)}^2}-\frac{1}{2^4(8n+7{)}^2}+\frac{1}{2(8n+1{)}^2})\\ & -2\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^{12n}}(\frac{1}{2^4(8n+2{)}^2}+\frac{1}{2^6(8n+3{)}^2}-\frac{1}{2^9(8n+5{)}^2}-\frac{1}{2^{10}(8n+6{)}^2}-\frac{1}{2^{12}(8n+7{)}^2}+\frac{1}{2^3(8n+1{)}^2})\end{array}$$ and $$G=\frac{\pi }{8}\log (2+\sqrt{3})+\frac{3}{8}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(2n+1{)}^2(\genfrac{}{}{0ex}{}{2n}{n})}.$$

The theoretical foundations for such series are given by Broadhurst, for the first formula, and Ramanujan, for the second formula. The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba. Using these series, calculating Catalan's constant is now about as fast as calculating Apéry's constant, $\zeta (3)$.

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

$G=\frac{1}{2}\sum _{k=0}^{\mathrm{\infty }}\frac{(-8{)}^k(3k+2)}{(2k+1{)}^3{(\genfrac{}{}{0ex}{}{2k}{k})}^3}$

$G=\frac{1}{64}\sum _{k=1}^{\mathrm{\infty }}\frac{256^k(580k^2-184k+15)}{k^3(2k-1)(\genfrac{}{}{0ex}{}{6k}{3k})(\genfrac{}{}{0ex}{}{6k}{4k})(\genfrac{}{}{0ex}{}{4k}{2k})}$

$G=-\frac{1}{1024}\sum _{k=1}^{\mathrm{\infty }}\frac{(-4096{)}^k(45136k^4-57184k^3+21240k^2-3160k+165)}{k^3(2k-1{)}^3}\left(\frac{(2k){!}^6(3k){!}^3}{k{!}^3(6k){!}^3}\right)$

All of these series have time complexity $O(n\log (n{)}^3)$.

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." Some of these expressions include: $$\begin{array}{rl}G& =-\frac{1}{\pi i}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\ln \ln \tan x\ln \tan xdx\\ G& =\underset{[0,1{]}^2}{\iint }\frac{1}{1+x^2{y}^2}dxdy\\ G& =\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1-x}{\int }}\frac{1}{1-x^2-y^2}dydx\\ G& =\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln t}{1+t^2}dt\\ G& =-\underset{0}{\overset{1}{\int }}\frac{\ln t}{1+t^2}dt\\ G& =\frac{1}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{t}{\sin t}dt\\ G& =\underset{0}{\overset{\frac{\pi }{4}}{\int }}\ln \cot tdt\\ G& =\frac{1}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\ln (\sec t+\tan t)dt\\ G& =\underset{0}{\overset{1}{\int }}\frac{\arccos t}{\sqrt{1+t^2}}dt\\ G& =\underset{0}{\overset{1}{\int }}\frac{\mathrm{arcsinh}t}{\sqrt{1-t^2}}dt\\ G& =\frac{1}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arctan t}{t\sqrt{1+t^2}}dt\\ G& =\frac{1}{2}\underset{0}{\overset{1}{\int }}\frac{\mathrm{arctanh}t}{\sqrt{1-t^2}}dt\\ G& =\underset{0}{\overset{\mathrm{\infty }}{\int }}\mathrm{arccot}{e}^{t}dt\\ G& =\frac{1}{4}\underset{0}{\overset{{\pi }^2/4}{\int }}\csc \sqrt{t}dt\\ G& =\frac{1}{16}({\pi }^2+4\underset{1}{\overset{\mathrm{\infty }}{\int }}{\mathrm{arccsc}}^{2}tdt)\\ G& =\frac{1}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{t}{\cosh t}dt\\ G& =\frac{\pi }{2}\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{(t^4-6t^2+1)\ln \ln t}{{(1+t^2)}^3}dt\\ G& =\frac{1}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\arcsin (\sin t)}{t}dt\\ G& =1+\underset{\alpha \to 1^{-}}{\lim }\{\underset{0}{\overset{\alpha }{\int }}\frac{(1+6t^2+t^4)\arctan t}{t{(1-t^2)}^2}dt+2\mathrm{artanh}\alpha -\frac{\pi \alpha }{1-{\alpha }^2}\}\\ G& =1-\frac{1}{8}\underset{{\mathbb{ℝ}}^2}{\iint }\frac{x\sin (2xy/\pi )}{(x^2+{\pi }^2)\cosh x\sinh y}dxdy\\ G& =\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sqrt[4]{x}(\sqrt{x}\sqrt{y}-1)}{(x+1{)}^2\sqrt[4]{y}(y+1{)}^2\log (xy)}dxdy\end{array}$$

where the last three formulas are related to Malmsten's integrals.

If Template:Math is the complete elliptic integral of the first kind, as a function of the elliptic modulus Template:Math, then $$G=\frac{1}{2}\underset{0}{\overset{1}{\int }}\mathrm{K}(k)dk$$

If Template:Math is the complete elliptic integral of the second kind, as a function of the elliptic modulus Template:Math, then $$G=-\frac{1}{2}+\underset{0}{\overset{1}{\int }}\mathrm{E}(k)dk$$

With the gamma function Template:Math $$\begin{array}{rl}G& =\frac{\pi }{4}\underset{0}{\overset{1}{\int }}\mathrm{\Gamma }(1+\frac{x}{2})\mathrm{\Gamma }(1-\frac{x}{2})dx\\ & =\frac{\pi }{2}\underset{0}{\overset{\frac{1}{2}}{\int }}\mathrm{\Gamma }(1+y)\mathrm{\Gamma }(1-y)dy\end{array}$$

The integral $$G={\mathrm{Ti}}_2(1)=\underset{0}{\overset{1}{\int }}\frac{\arctan t}{t}dt$$ 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:

$$\begin{array}{rl}{\psi }_1\left(\frac{1}{4}\right)& ={\pi }^2+8G\\ {\psi }_1\left(\frac{3}{4}\right)& ={\pi }^2-8G.\end{array}$$

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 (see External links below).

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):

$$G=4\pi \log \left(\frac{G\left(\frac{3}{8}\right)G\left(\frac{7}{8}\right)}{G\left(\frac{1}{8}\right)G\left(\frac{5}{8}\right)}\right)+4\pi \log \left(\frac{\mathrm{\Gamma }\left(\frac{3}{8}\right)}{\mathrm{\Gamma }\left(\frac{1}{8}\right)}\right)+\frac{\pi }{2}\log \left(\frac{1+\sqrt{2}}{2(2-\sqrt{2})}\right).$$

If one defines the Lerch transcendent Template:Math by $$\mathrm{\Phi }(z,s,\alpha )=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{(n+\alpha {)}^s},$$ then $$G=\frac{1}{4}\mathrm{\Phi }(-1,2,\frac{1}{2}).$$

Continued fraction

Template:Mvar can be expressed in the following form:

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

The simple continued fraction is given by:

$G=\frac{1}{1+\frac{1}{10+\frac{1}{1+\frac{1}{8+\frac{1}{1+\frac{1}{88+\ddots }}}}}}$

This continued fraction would have infinite terms if and only if $G$ is irrational, which is still unresolved. The following continued fraction representation gives (asymptotically) 2.08 new correct decimal places per cycle:

$G=\frac{\frac{13}{2}}{Z_0},Z_k=a(k)+\frac{b(k)}{Z_{k+1}}$

with

$a(k)=3520k^6+5632k^5+2064k^4-384k^3-156k^2+16k+7$

$b(k)=(2k+1{)}^4(2k+2{)}^4(20k^2-8k+1)(20k^2+72k+65)$

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.

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	Shigeru Kondo & Steve Pagliarulo
August 2008	Template:Val	Shigeru Kondo & Steve Pagliarulo
January 31, 2009	Template:Val	Alexander J. Yee & Raymond Chan
April 16, 2009	Template:Val	Alexander J. Yee & Raymond Chan
June 7, 2015	Template:Val	Robert J. Setti
April 12, 2016	Template:Val	Ron Watkins
February 16, 2019	Template:Val	Tizian Hanselmann
March 29, 2019	Template:Val	Mike A & Ian Cutress
July 16, 2019	Template:Val	Seungmin Kim
September 6, 2020	Template:Val	Andrew Sun
March 9, 2022	Template:Val	Seungmin Kim

See also

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

External links

• (Provides over one hundred different identities).
• (Provides a graphical interpretation of the relations)
• (Provides the first 300,000 digits of Catalan's constant)
• Template:Springer