# Catalan's constant

In mathematics, Catalan's constant G, which occasionally appears in estimates in combinatorics, is defined by

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

where β is the Dirichlet beta function. Its numerical value[1] is approximately (sequence A006752 in the OEIS)

G = 0.915965594177219015054603514932384110774
 Unsolved problem in mathematics: Is Catalan's constant irrational? If so, is it transcendental? (more unsolved problems in mathematics)

It is not known whether G is irrational, let alone transcendental.[2]

Catalan's constant was named after Eugène Charles Catalan.

The similar but apparently more complicated series

${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{3}}}={\frac {1}{1^{3}}}-{\frac {1}{3^{3}}}+{\frac {1}{5^{3}}}-{\frac {1}{7^{3}}}+\cdots \!}$

can be evaluated exactly and is π3/32.

## Integral identities

Some identities involving definite integrals include

{\displaystyle {\begin{aligned}G&=\int _{0}^{1}\int _{0}^{1}{\frac {1}{1+x^{2}y^{2}}}\,dx\,dy\\G&=-\int _{0}^{1}{\frac {\ln t}{1+t^{2}}}\,dt\\G&=\int _{0}^{\frac {\pi }{4}}{\frac {t}{\sin t\cos t}}\,dt\\G&={\tfrac {1}{4}}\int _{-{\frac {\pi }{2}}}^{\frac {\pi }{2}}{\frac {t}{\sin t}}\,dt\\G&=\int _{0}^{\frac {\pi }{4}}\ln \cot t\,dt\\G&={\tfrac {1}{2}}\int _{0}^{\frac {\pi }{2}}\ln(\sec t+\tan t)\,dt\\G&=\int _{0}^{\infty }\arctan e^{-t}\,dt\\G&=\int _{1}^{\infty }{\frac {\ln t}{1+t^{2}}}\,dt\\G&={\tfrac {1}{2}}\int _{0}^{\infty }{\frac {t}{\cosh t}}\,dt\end{aligned}}}

If K(t) is a complete elliptic integral of the first kind, then

${\displaystyle G={\tfrac {1}{2}}\int _{0}^{1}\mathrm {K} (t)\,dt}$

With the gamma function Γ(x + 1) = x!

{\displaystyle {\begin{aligned}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}^{\frac {1}{2}}\Gamma (1+y)\Gamma (1-y)\,dy\end{aligned}}}

The integral

${\displaystyle G=\operatorname {Ti} _{2}(1)=\int _{0}^{1}{\frac {\arctan t}{t}}\,dt}$

is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.

## Uses

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

{\displaystyle {\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\\\psi _{1}\left({\tfrac {3}{4}}\right)&=\pi ^{2}-8G.\end{aligned}}}

Simon Plouffe gives an infinite collection of identities between the trigamma function, π2 and Catalan's constant; these are expressible as paths on a graph.

In low-dimensional topology, Catalan's constant is a rational multiple of the volume of an ideal hyperbolic octahedron, and therefore of the hyperbolic volume of the complement of the Whitehead link.[3]

It also appears in connection with the hyperbolic secant distribution.

## Relation to other special functions

Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes G-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 G-function, the following expression is easily obtained (see Clausen function for more):

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

If one defines the Lerch transcendent Φ(z,s,α) (related to the Lerch zeta function) by

${\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}},}$

then it is clear that

${\displaystyle G={\tfrac {1}{4}}\Phi \left(-1,2,{\tfrac {1}{2}}\right).}$

## Quickly converging series

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

{\displaystyle {\begin{aligned}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{aligned}}}

and

${\displaystyle G={\frac {\pi }{8}}\log \left(2+{\sqrt {3}}\right)+{\frac {3}{8}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}{\binom {2n}{n}}}}.}$

The theoretical foundations for such series are given by Broadhurst, for the first formula,[4] and Ramanujan, for the second formula.[5] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.[6][7]

## Known digits

The number of known digits of Catalan's constant G has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.[8]

Number of known decimal digits of Catalan's constant G
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 20000 Greg J. Fee
1996 50000 Greg J. Fee
August 14, 1996 100000 Greg J. Fee & Simon Plouffe
September 29, 1996 300000 Thomas Papanikolaou
1996 1500000 Thomas Papanikolaou
1997 3379957 Patrick Demichel
January 4, 1998 12500000 Xavier Gourdon
2001 100000500 Xavier Gourdon & Pascal Sebah
2002 201000000 Xavier Gourdon & Pascal Sebah
October 2006 5000000000 Shigeru Kondo & Steve Pagliarulo[9]
August 2008 10000000000 Shigeru Kondo & Steve Pagliarulo[10]
January 31, 2009 15510000000 Alexander J. Yee & Raymond Chan[11]
April 16, 2009 31026000000 Alexander J. Yee & Raymond Chan[11]
April 6, 2013 100000000000 Robert J. Setti
June 7, 2015 200000001100 Robert J. Setti[12]

## Notes

1. ^ [1]
2. ^ Nesterenko, Yu. V. (January 2016), "On Catalan's constant", Proceedings of the Steklov Institute of Mathematics, 292 (1): 153–170, doi:10.1134/s0081543816010107.
3. ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, doi:10.1090/S0002-9939-10-10364-5, MR 2661571.
4. ^ Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:.
5. ^ Berndt, B. C. (1985). Ramanujan's Notebook, Part I. Springer Verlag.[ISBN missing]
6. ^ Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (4): 339–360. MR 1156939. Zbl 0754.65021.
7. ^ Karatsuba, E. A. (2001). "Fast computation of some special integrals of mathematical physics". In Krämer, W.; von Gudenberg, J. W. Scientific Computing, Validated Numerics, Interval Methods. p. 29–41.[ISBN missing]
8. ^ Gourdon, X.; Sebah, P. "Constants and Records of Computation".
9. ^ Shigeru Kondo's website
10. ^ Constants and Records of Computation
11. ^ a b Large Computations
12. ^ Catalan's constant records using YMP