# Catalan's constant

In mathematics, Catalan's constant G, 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?

It is not known whether G is irrational, let alone transcendental.[2] G has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven".[3]

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.[4][5]

## 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.[6] It is 1/8 of the volume of the complement of the Borromean rings.[7]

In combinatorics and statistical mechanics, it arises in connection with counting domino tilings,[8] spanning trees,[9] and Hamiltonian cycles of grid graphs.[10]

In number theory, Catalan's constant appears in a conjectured formula for the asymptotic number of primes of the form ${\displaystyle n^{2}+1}$ 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.[11]

Catalan's constant also appears in the calculation of the mass distribution of spiral galaxies.[12][13]

## 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.[14]

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[15]
August 2008 10000000000 Shigeru Kondo & Steve Pagliarulo[14]
January 31, 2009 15510000000 Alexander J. Yee & Raymond Chan[16]
April 16, 2009 31026000000 Alexander J. Yee & Raymond Chan[16]
June 7, 2015 200000001100 Robert J. Setti[17]
April 12, 2016 250000000000 Ron Watkins[17]
February 16, 2019 300000000000 Tizian Hanselmann[17]
March 29, 2019 500000000000 Mike A & Ian Cutress[17]
July 16, 2019 600000000100 Seungmin Kim[18][19]
September 6, 2020 1000000001337 Andrew Sun[20]
March 9, 2022 1200000000100 Seungmin Kim[20]

## 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."[21] Some of these expressions include:

{\displaystyle {\begin{aligned}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 }\operatorname {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 }\operatorname {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-{\frac {1}{8}}\iint _{\mathbb {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(xy)}}dxdy\end{aligned}}}

where the last three formulas are related to Malmsten's integrals.[22]

If K(k) is the complete elliptic integral of the first kind, as a function of the elliptic modulus k, then

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

If E(k) is the complete elliptic integral of the second kind, as a function of the elliptic modulus k, then

${\displaystyle G=-{\tfrac {1}{2}}+\int _{0}^{1}\mathrm {E} (k)\,dk}$

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.

## Relation to other special functions

G appears 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.

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

## 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,[23] and Ramanujan, for the second formula.[24] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.[25][26] Using these series, calculating Catalan's constant is now about as fast as calculating Apery's constant, ${\displaystyle \zeta (3)}$.[27]

Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:[27]

${\displaystyle G={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {(-8)^{k}(3k+2)}{(2k+1)^{3}{\binom {2k}{k}}^{3}}}}$
${\displaystyle G={\frac {1}{64}}\sum _{k=1}^{\infty }{\frac {256^{k}(580k^{2}-184k+15)}{k^{3}(2k-1){\binom {6k}{3k}}{\binom {6k}{4k}}{\binom {4k}{2k}}}}}$
${\displaystyle G=-{\frac {1}{1024}}\sum _{k=1}^{\infty }{\frac {(-4096)^{k}(45136k^{4}-57184k^{3}+21240k^{2}-3160k+165)}{k^{3}(2k-1)^{3}}}\left({\frac {(2k)!^{6}(3k)!^{3}}{k!^{3}(6k)!^{3}}}\right)}$

All of these series have time complexity ${\displaystyle O(n\log(n)^{3})}$.[27]

## Continued fraction

G can be expressed in the following form[28]

${\displaystyle G={\cfrac {1}{1+{\cfrac {1^{4}}{8+{\cfrac {3^{4}}{16+{\cfrac {5^{4}}{24+{\cfrac {7^{4}}{32+{\cfrac {9^{4}}{40+\ddots }}}}}}}}}}}}}$
The simple continued fraction is given by[29]
${\displaystyle G={\cfrac {1}{1+{\cfrac {1}{10+{\cfrac {1}{1+{\cfrac {1}{8+{\cfrac {1}{1+{\cfrac {1}{88+\ddots }}}}}}}}}}}}}$
This continued fraction would have infinite terms if and only if ${\displaystyle G}$ is irrational, which is still unresolved.

## References

1. ^ Papanikolaou, Thomas (March 1997). Catalan's Constant to 1,500,000 Places – via Gutenberg.org.
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, S2CID 124903059.
3. ^ Bailey, David H.; Borwein, Jonathan M.; Mattingly, Andrew; Wightwick, Glenn (2013), "The computation of previously inaccessible digits of ${\displaystyle \pi ^{2}}$ and Catalan's constant", Notices of the American Mathematical Society, 60 (7): 844–854, doi:10.1090/noti1015, MR 3086394
4. ^ Goldstein, Catherine (2015), "The mathematical achievements of Eugène Catalan", Bulletin de la Société Royale des Sciences de Liège, 84: 74–92, MR 3498215
5. ^ Catalan, E. (1865), "Mémoire sur la transformation des séries et sur quelques intégrales définies", Ers, Publiés Par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. Collection in 4, Mémoires de l'Académie royale des sciences, des lettres et des beaux-arts de Belgique (in French), 33, Brussels, hdl:2268/193841
6. ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002-9939-10-10364-5, MR 2661571, S2CID 2016662.
7. ^ William Thurston (March 2002), "7. Computation of volume" (PDF), The Geometry and Topology of Three-Manifolds, p. 165, archived (PDF) from the original on 2011-01-25
8. ^ Temperley, H. N. V.; Fisher, Michael E. (August 1961), "Dimer problem in statistical mechanics—an exact result", Philosophical Magazine, 6 (68): 1061–1063, Bibcode:1961PMag....6.1061T, doi:10.1080/14786436108243366
9. ^ Wu, F. Y. (1977), "Number of spanning trees on a lattice", Journal of Physics, 10 (6): L113–L115, Bibcode:1977JPhA...10L.113W, doi:10.1088/0305-4470/10/6/004, MR 0489559
10. ^ Kasteleyn, P. W. (1963), "A soluble self-avoiding walk problem", Physica, 29 (12): 1329–1337, Bibcode:1963Phy....29.1329K, doi:10.1016/S0031-8914(63)80241-4, MR 0159642
11. ^ Shanks, Daniel (1959), "A sieve method for factoring numbers of the form ${\displaystyle n^{2}+1}$", Mathematical Tables and Other Aids to Computation, 13: 78–86, doi:10.2307/2001956, JSTOR 2001956, MR 0105784
12. ^ Wyse, A. B.; Mayall, N. U. (January 1942), "Distribution of Mass in the Spiral Nebulae Messier 31 and Messier 33.", The Astrophysical Journal, 95: 24–47, Bibcode:1942ApJ....95...24W, doi:10.1086/144370
13. ^ van der Kruit, P. C. (March 1988), "The three-dimensional distribution of light and mass in disks of spiral galaxies.", Astronomy & Astrophysics, 192: 117–127, Bibcode:1988A&A...192..117V
14. ^ a b Gourdon, X.; Sebah, P. "Constants and Records of Computation". Retrieved 11 September 2007.
15. ^ "Shigeru Kondo's website". Archived from the original on 2008-02-11. Retrieved 2008-01-31.
16. ^ a b "Large Computations". Retrieved 31 January 2009.
17. ^ a b c d "Catalan's constant records using YMP". Retrieved 14 May 2016.
18. ^ "Catalan's constant records using YMP". Archived from the original on 22 July 2019. Retrieved 22 July 2019.
19. ^ "Catalan's constant world record by Seungmin Kim". 23 July 2019. Retrieved 17 October 2020.
20. ^ a b "Records set by y-cruncher". www.numberworld.org. Retrieved 2022-02-13.
21. ^ Stewart, Seán M. (2020), "A Catalan constant inspired integral odyssey", The Mathematical Gazette, 104 (561): 449–459, doi:10.1017/mag.2020.99, MR 4163926, S2CID 225116026
22. ^ Blagouchine, Iaroslav (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results" (PDF). The Ramanujan Journal. 35: 21–110. doi:10.1007/s11139-013-9528-5. S2CID 120943474. Archived from the original (PDF) on 2018-10-02. Retrieved 2018-10-01.
23. ^ Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:math.CA/9803067.
24. ^ Berndt, B. C. (1985). Ramanujan's Notebook, Part I. Springer Verlag. p. 289. ISBN 978-1-4612-1088-7.
25. ^ Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (4): 339–360. MR 1156939. Zbl 0754.65021.
26. ^ Karatsuba, E. A. (2001). "Fast computation of some special integrals of mathematical physics". In Krämer, W.; von Gudenberg, J. W. (eds.). Scientific Computing, Validated Numerics, Interval Methods. pp. 29–41. doi:10.1007/978-1-4757-6484-0_3.
27. ^ a b c Alexander Yee (14 May 2019). "Formulas and Algorithms". Retrieved 5 December 2021.
28. ^ Bowman, D. & Mc Laughlin, J. (2002). "Polynomial continued fractions" (PDF). Acta Arithmetica. 103 (4): 329–342. arXiv:1812.08251. Bibcode:2002AcAri.103..329B. doi:10.4064/aa103-4-3. S2CID 119137246. Archived (PDF) from the original on 2020-04-13.
29. ^ "A014538 - OEIS". oeis.org. Retrieved 2022-10-27.