Editing Catalan's constant

{{short description|Number, approximately 0.916}}
{{Distinguish|Catalan number}}
{{infobox non-integer number
| rationality = Unknown
| symbol = ''G''
| decimal = {{gaps|0.91596|55941|77219|0150...}}
}}
[[File:Catalan constant area.png|thumb|catalan constant as area under the curve of arctanx /x]]
{{CS1 config|mode=cs1}}
In [[mathematics]], '''Catalan's constant''' {{mvar|G}}, is the alternating sum of the reciprocals of the odd [[Square number|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 {{mvar|β}} is the [[Dirichlet beta function]]. Its numerical value<ref>{{cite book|last1=Papanikolaou|first1=Thomas| title=Catalan's Constant to 1,500,000 Places|url=https://www.gutenberg.org/ebooks/812|via=Gutenberg.org|date=March 1997}}</ref> is approximately {{OEIS|A006752}}

: {{math|1=''G'' = {{val|0.915965594177219015054603514932384110774}}…}}

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>{{citation
 | last = Goldstein | first = Catherine | author-link = Catherine Goldstein
 | journal = Bulletin de la Société Royale des Sciences de Liège
 | mr = 3498215
 | pages = 74–92
 | title = The mathematical achievements of Eugène Catalan
 | url = https://popups.uliege.be/0037-9565/index.php?id=4830
 | volume = 84
 | year = 2015}}</ref><ref>{{citation
 | last = Catalan | first = E. | author-link = Eugène Charles Catalan
 | hdl = 2268/193841
 | language = fr
 | location = Brussels
 | series = Mémoires de l'Académie royale des sciences, des lettres et des beaux-arts de Belgique
 | title = Mémoire sur la transformation des séries et sur quelques intégrales définies
 | journal = Ers, Publiés Par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique. Collection in 4 | volume = 33
 | year = 1865}}</ref>

==Uses==
In [[low-dimensional topology]], Catalan's constant is 1/4 of the volume of an [[ideal polyhedron|ideal]] hyperbolic [[octahedron]], and therefore 1/4 of the [[hyperbolic volume]] of the complement of the [[Whitehead link]].<ref>{{citation
 | last = Agol | first = Ian | author-link = Ian Agol
 | doi = 10.1090/S0002-9939-10-10364-5
 | issue = 10
 | journal = [[Proceedings of the American Mathematical Society]]
 | mr = 2661571
 | pages = 3723–3732
 | title = The minimal volume orientable hyperbolic 2-cusped 3-manifolds
 | volume = 138
 | year = 2010| arxiv = 0804.0043| s2cid = 2016662 }}.</ref> It is 1/8 of the volume of the complement of the [[Borromean rings]].<ref>{{Citation |author=William Thurston|author-link=William Thurston |date=March 2002 |title=The Geometry and Topology of Three-Manifolds |url=http://library.msri.org/books/gt3m/ |chapter=7. Computation of volume |chapter-url=http://library.msri.org/books/gt3m/PDF/7.pdf |archive-url=https://web.archive.org/web/20110125012649/http://library.msri.org/books/gt3m/PDF/7.pdf |archive-date=2011-01-25 |url-status=live |page=165}}</ref>

In [[combinatorics]] and [[statistical mechanics]], it arises in connection with counting [[domino tiling]]s,<ref>{{citation
 | last1 = Temperley | first1 = H. N. V. | author1-link = Harold Neville Vazeille Temperley
 | last2 = Fisher | first2 = Michael E. | author2-link = Michael Fisher
 | date = August 1961
 | doi = 10.1080/14786436108243366
 | issue = 68
 | journal = [[Philosophical Magazine]]
 | pages = 1061–1063
 | title = Dimer problem in statistical mechanics—an exact result
 | volume = 6| bibcode = 1961PMag....6.1061T }}</ref> [[spanning tree]]s,<ref>{{citation
 | last = Wu | first = F. Y.
 | doi = 10.1088/0305-4470/10/6/004
 | issue = 6
 | journal = Journal of Physics
 | mr = 489559
 | pages = L113–L115
 | title = Number of spanning trees on a lattice
 | volume = 10
 | year = 1977| bibcode = 1977JPhA...10L.113W
 }}</ref> and [[Hamiltonian cycle]]s of [[grid graph]]s.<ref>{{citation
 | last = Kasteleyn | first = P. W. | author-link = Pieter Kasteleyn
 | doi = 10.1016/S0031-8914(63)80241-4
 | journal = [[Physica (journal)|Physica]]
 | mr = 159642
 | pages = 1329–1337
 | title = A soluble self-avoiding walk problem
 | volume = 29
 | year = 1963| issue = 12 | bibcode = 1963Phy....29.1329K }}</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 [[Ulam spiral#Hardy and Littlewood's Conjecture F|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>{{citation
 | last = Shanks | first = Daniel | author-link = Daniel Shanks
 | journal = Mathematical Tables and Other Aids to Computation
 | mr = 105784
 | pages = 78–86
 | title = A sieve method for factoring numbers of the form <math>n^2+1</math>
 | volume = 13
 | year = 1959| doi = 10.2307/2001956 | jstor = 2001956 }}</ref>

Catalan's constant also appears in the calculation of the [[mass distribution]] of [[spiral galaxy|spiral galaxies]].<ref>{{citation
 | last1 = Wyse | first1 = A. B. | author1-link=Arthur Bambridge Wyse
 | last2 = Mayall | first2 = N. U.
 | bibcode = 1942ApJ....95...24W
 | date = January 1942
 | doi = 10.1086/144370
 | journal = [[The Astrophysical Journal]]
 | pages = 24–47
 | title = Distribution of Mass in the Spiral Nebulae Messier 31 and Messier 33.
 | volume = 95| doi-access = free
 }}</ref><ref>{{citation
 | last = van der Kruit | first = P. C.
 | bibcode = 1988A&A...192..117V
 | date = March 1988
 | journal = [[Astronomy & Astrophysics]]
 | pages = 117–127
 | title = The three-dimensional distribution of light and mass in disks of spiral galaxies.
 | volume = 192}}</ref>

==Properties==

{{unsolved|mathematics|Is Catalan's constant irrational? If so, is it transcendental?}}
It is not known whether {{mvar|G}} is [[irrational number|irrational]], let alone [[transcendental number|transcendental]].<ref>{{citation
 | last = Nesterenko | first = Yu. V.
 | date = January 2016
 | doi = 10.1134/s0081543816010107
 | issue = 1
 | journal = Proceedings of the Steklov Institute of Mathematics
 | pages = 153–170
 | title = On Catalan's constant
 | volume = 292| s2cid = 124903059
 }}.</ref> {{mvar|G}} has been called "arguably the most basic constant whose irrationality and transcendence (though strongly
suspected) remain unproven".<ref>{{citation
 | last1 = Bailey | first1 = David H.
 | last2 = Borwein | first2 = Jonathan M.
 | last3 = Mattingly | first3 = Andrew
 | last4 = Wightwick | first4 = Glenn
 | doi = 10.1090/noti1015
 | issue = 7
 | journal = [[Notices of the American Mathematical Society]]
 | mr = 3086394
 | pages = 844–854
 | title = The computation of previously inaccessible digits of <math>\pi^2</math> and Catalan's constant
 | volume = 60
 | year = 2013| doi-access = free
 }}</ref>

There exist however partial results. It is known that infinitely many of the numbers ''β''(2''n'') are irrational, where ''β(s)'' is the Dirichlet beta function.<ref>{{Cite journal |last1=Rivoal |first1=T. |last2=Zudilin |first2=W. |date=2003-08-01 |title=Diophantine properties of numbers related to Catalan's constant |url=https://link.springer.com/article/10.1007/s00208-003-0420-2 |journal=Mathematische Annalen |language=en |volume=326 |issue=4 |pages=705–721 |doi=10.1007/s00208-003-0420-2 |hdl=1959.13/803688 |issn=1432-1807}}</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>{{Cite journal |last=Zudilin |first=Wadim |date=2018-04-26 |title=Arithmetic of Catalan's constant and its relatives |journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |volume=89 |pages=45–53 |doi=10.1007/s12188-019-00203-w |arxiv=1804.09922 |language=en}}</ref> These results by [[Wadim Zudilin]] and [[Tanguy Rivoal]] are related to similar ones given for the [[Particular values of the Riemann zeta function#Odd positive integers|odd zeta constants]] ζ(2''n+1'').

Catalan's constant is known to be an [[Period (algebraic geometry)|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">{{Cite web |last=Weisstein |first=Eric W. |title=Catalan's Constant |url=https://mathworld.wolfram.com/CatalansConstant.html |access-date=2024-10-02 |website=mathworld.wolfram.com |language=en}}</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>{{cite arXiv|first1=D. J. |last1=Broadhurst|eprint=math.CA/9803067 |title=Polylogarithmic ladders, hypergeometric series and the ten millionth digits of {{math|''ζ''(3)}} and {{math|''ζ''(5)}}| year=1998}}</ref> and Ramanujan, for the second formula.<ref>{{cite book|first=B. C.|last=Berndt|title=Ramanujan's Notebook, Part I|publisher=Springer Verlag|date=1985|page=289|isbn=978-1-4612-1088-7}}</ref> The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.<ref>{{cite journal|first=E. A.| last=Karatsuba| title=Fast evaluation of transcendental functions|journal=Probl. Inf. Transm.|volume=27|issue=4| pages=339–360| date=1991|zbl=0754.65021|mr=1156939}}</ref><ref>{{cite book|first=E. A.|last=Karatsuba|contribution=Fast computation of some special integrals of mathematical physics|title=Scientific Computing, Validated Numerics, Interval Methods| url=https://archive.org/details/scientificcomput00wals_919|url-access=limited|editor1-first=W.|editor1-last=Krämer| editor2-first=J. W.|editor2-last=von Gudenberg|pages=[https://archive.org/details/scientificcomput00wals_919/page/n35 29]–41|date=2001|doi=10.1007/978-1-4757-6484-0_3|isbn=978-1-4419-3376-8 }}</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">{{cite web|url=http://www.numberworld.org/y-cruncher/internals/formulas.html|title=Formulas and Algorithms|author=Alexander Yee|date=14 May 2019|access-date=5 December 2021}}</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>{{citation | last = Stewart | first = Seán M. | doi = 10.1017/mag.2020.99 | issue = 561 | journal = [[The Mathematical Gazette]] | mr = 4163926 | pages = 449–459 | title = A Catalan constant inspired integral odyssey | volume = 104 | year = 2020| s2cid = 225116026 }}</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 [[Carl Johan Malmsten|Malmsten's]] integrals.<ref>{{Cite journal| first1=Iaroslav| last1=Blagouchine| title=Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results| year=2014| doi=10.1007/s11139-013-9528-5| url=https://iblagouchine.perso.centrale-marseille.fr/publications/Blagouchine-Malmsten-integrals-and-their-evaluation-by-contour-integration-methods-(Ramanujan-J-2014).pdf| volume=35| journal=The Ramanujan Journal| pages=21–110| s2cid=120943474| access-date=2018-10-01| archive-url=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| archive-date=2018-10-02| url-status=dead }}</ref>

If {{math|K(''k'')}} is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]], as a function of the elliptic modulus {{math|''k''}}, then
<math display="block"> G = \tfrac{1}{2} \int_0^1 \mathrm{K}(k)\,dk </math>

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

With the [[gamma function]] {{math|1=Γ(''x'' + 1) = ''x''!}}
<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==
{{mvar|G}} 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, {{pi}}<sup>2</sup> 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 {{mvar|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 {{mvar|G}}-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]]''' {{math|Φ(''z'',''s'',''α'')}} 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==
{{mvar|G}} can be expressed in the following form:<ref>{{cite journal | journal=Acta Arithmetica |volume=103 |issue=4 |pages=329–342 | author=Bowman, D. | author2=Mc Laughlin, J.| name-list-style=amp | title=Polynomial continued fractions | language=English | year=2002 |doi=10.4064/aa103-4-3 |arxiv=1812.08251 |bibcode=2002AcAri.103..329B |s2cid=119137246 | url=https://www.wcupa.edu/sciences-mathematics/mathematics/jMcLaughlin/documents/4paper1.pdf |archive-url=https://web.archive.org/web/20200413012537/https://www.wcupa.edu/sciences-mathematics/mathematics/jMcLaughlin/documents/4paper1.pdf |archive-date=2020-04-13 |url-status=live}}</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>{{Cite web |title=A014538 - OEIS |url=http://oeis.org/A014538 |access-date=2022-10-27 |website=oeis.org}}</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 {{mvar|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.<ref name=Gourdon>{{cite web|last1=Gourdon|first1=X.|last2=Sebah|first2=P.|url=http://numbers.computation.free.fr/Constants/constants.html|title=Constants and Records of Computation|access-date=11 September 2007}}</ref>

{| class="wikitable" style="margin: 1em auto 1em auto"
|+ Number of known decimal digits of Catalan's constant {{mvar|G}}
! Date || Decimal digits || Computation performed by
|-
| 1832 ||align="right"| 16 || [[Thomas Clausen (mathematician)|Thomas Clausen]]
|-
| 1858 ||align="right"| 19 || Carl Johan Danielsson Hill
|-
| 1864 ||align="right"| 14 || [[Eugène Charles Catalan]]
|-
| 1877 ||align="right"| 20 || [[James Whitbread Lee Glaisher|James W. L. Glaisher]]
|-
| 1913 ||align="right"| 32 || [[James Whitbread Lee Glaisher|James W. L. Glaisher]]
|-
| 1990 ||align="right"| {{val|20000}} || Greg J. Fee
|-
| 1996 ||align="right"| {{val|50000}} || Greg J. Fee
|-
| August 14, 1996 ||align="right"| {{val|100000}} || Greg J. Fee & [[Simon Plouffe]]
|-
| September 29, 1996 ||align="right"| {{val|300000}} || Thomas Papanikolaou
|-
| 1996 ||align="right"| {{val|1500000}} || Thomas Papanikolaou
|-
| 1997 ||align="right"| {{val|3379957}} || Patrick Demichel
|-
| January 4, 1998 ||align="right"| {{val|12500000}} || Xavier Gourdon
|-
| 2001 ||align="right"| {{val|100000500}} || Xavier Gourdon & Pascal Sebah
|-
| 2002 ||align="right"| {{val|201000000}} || Xavier Gourdon & Pascal Sebah
|-
| October 2006 ||align="right"| {{val|5000000000}} || Shigeru Kondo & Steve Pagliarulo<ref>{{Cite web |url=http://ja0hxv.calico.jp/pai/ecatalan.html |title=Shigeru Kondo's website |access-date=2008-01-31 |archive-url=https://web.archive.org/web/20080211185703/http://ja0hxv.calico.jp/pai/ecatalan.html |archive-date=2008-02-11 |url-status=dead }}</ref>
|-
| August 2008 ||align="right"| {{val|10000000000}} || Shigeru Kondo & Steve Pagliarulo<ref name=Gourdon />
|-
| January 31, 2009 ||align="right"| {{val|15510000000}} || Alexander J. Yee & Raymond Chan<ref name=yee_chan>{{cite web| url = http://www.numberworld.org/nagisa_runs/computations.html| title = Large Computations |accessdate=31 January 2009}}</ref>
|-
| April 16, 2009 ||align="right"| {{val|31026000000}} || Alexander J. Yee & Raymond Chan<ref name=yee_chan/>
|-
| June 7, 2015 ||align="right"| {{val|200000001100}} || Robert J. Setti<ref name=setti>{{cite web| url = http://www.numberworld.org/digits/Catalan/| title = Catalan's constant records using YMP |access-date=14 May 2016}}</ref>
|-
| April 12, 2016 ||align="right"| {{val|250000000000}} || Ron Watkins<ref name=setti/>
|-
| February 16, 2019  ||align="right"| {{val|300000000000}} || Tizian Hanselmann<ref name=setti/>
|-
| March 29, 2019  ||align="right"| {{val|500000000000}} || Mike A & Ian Cutress<ref name=setti/>
|-
| July 16, 2019  ||align="right"| {{val|600000000100}} || Seungmin Kim<ref name=yee>{{cite web |url = http://www.numberworld.org/y-cruncher/ |title = Catalan's constant records using YMP |archive-url=https://web.archive.org/web/20190722034426/http://www.numberworld.org/y-cruncher/ |archive-date=22 July 2019 |url-status=dead |access-date=22 July 2019}}</ref><ref name=kim>{{cite web| url = https://ehfd.github.io/world-record/catalans-constant/| title = Catalan's constant world record by Seungmin Kim| date = 23 July 2019 |access-date=17 October 2020}}</ref>
|-
| September 6, 2020  ||align="right"| {{val|1000000001337}} || Andrew Sun<ref name=yrecord>{{Cite web|title=Records set by y-cruncher|url=http://www.numberworld.org/y-cruncher/records.html|access-date=2022-02-13|website=www.numberworld.org}}</ref>
|-
| March 9, 2022 ||align="right"| {{val|1200000000100}} || Seungmin Kim<ref name=yrecord/>
|-
|}

==See also ==
* [[Gieseking manifold]]
* [[List of mathematical constants]]
* [[Mathematical constant]]
* [[Particular values of Riemann zeta function]]

==References==
{{reflist}}

==Further reading==
* {{cite journal
 |first        = Victor
 |last         = Adamchik
 |year         = 2002
 |journal      = Zeitschrift für Analysis und ihre Anwendungen
 |volume       = 21
 |issue        = 3
 |pages        = 1–10
 |title        = A certain series associated with Catalan's constant
 |mr           = 1929434
 |doi          = 10.4171/ZAA/1110
 |doi-access   = free
 }}
* {{cite conference
 | last = Fee | first = Gregory J.
 | editor1-last = Watanabe | editor1-first = Shunro
 | editor2-last = Nagata | editor2-first = Morio
 | contribution = Computation of Catalan's Constant Using Ramanujan's Formula
 | doi = 10.1145/96877.96917
 | isbn    = 0201548925
 | s2cid   = 1949187
 | pages = 157–160
 | publisher = ACM
 | title = Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC '90, Tokyo, Japan, August 20-24, 1990
 | year = 1990| doi-access = free
 }}
* {{Cite journal
  | first   = David M.
  | last    = Bradley
  | title   = A class of series acceleration formulae for Catalan's constant
  | doi     = 10.1023/A:1006945407723
  | year    = 1999
  | journal = The Ramanujan Journal
  | volume  = 3
  | issue   = 2
  | pages   = 159–173
  | mr      = 1703281
  | arxiv   = 0706.0356
  | bibcode = 2007arXiv0706.0356B
  | s2cid   = 5111792
  }}

==External links==
* {{cite web
  | first       = Victor
  | last        = Adamchik
  | url         = http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
  | title       = 33 representations for Catalan's constant
  | archiveurl  = https://web.archive.org/web/20160807111945/https://www.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
  | archivedate = 2016-08-07
  | access-date = 14 July 2005
  }}
* {{cite web
  | first       = Simon
  | last        = Plouffe
  | url         = http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html
  | title       = A few identities (III) with Catalan
  | year        = 1993
  | archiveurl  = https://web.archive.org/web/20190626124124/https://lacim.uqam.ca/~plouffe/IntegerRelations/identities3a.html
  | archivedate = 2019-06-26
  | access-date = 29 July 2005
  }} (Provides over one hundred different identities).
* {{cite web
  | first       = Simon
  | last        = Plouffe
  | url         = http://www.lacim.uqam.ca/~plouffe/IntegerRelations/identities3.html
  | title       = A few identities with Catalan constant and Pi^2
  | year        = 1999
  | archiveurl  = https://web.archive.org/web/20190626124128/https://lacim.uqam.ca/~plouffe/IntegerRelations/identities3.html
  | archivedate = 2019-06-26
  | access-date  = 29 July 2005
  }} (Provides a graphical interpretation of the relations)
* {{cite book
  | first  = Greg
  | last   = Fee
  | url    = https://www.gutenberg.org/ebooks/682
  | title  = Catalan's Constant (Ramanujan's Formula)
  | year   = 1996
  }} (Provides the first 300,000 digits of Catalan's constant)
* {{citation
  | first     = David M.
  | last      = Bradley
  | title     = Representations of Catalan's constant
  | citeseerx = 10.1.1.26.1879
  | year      = 2001
  | url = https://www.researchgate.net/publication/2325473
  | mode = cs1
  }}
* {{cite web | first = Fredrik | last = Johansson |url = https://fungrim.org/ordner/0.915965594177219015054603514932/
 | title = 0.915965594177219015054603514932 | work = Ordner, a catalog of real numbers in Fungrim | accessdate= 21 April 2021 }}
* {{cite web
  | title     = Catalan's Constant
  | url       = https://www.youtube.com/watch?v=e5wqw2_EkxQ&list=PLW1_9UnhaSkGqlwbQphLMGCx2JvaGu1HB&index=72
  | publisher = Let's Learn, Nemo!
  | date      = 10 August 2020
  | website   = [[YouTube]]
  | access-date= 6 April 2021
  }}
* {{MathWorld |title = Catalan's Constant |urlname = CatalansConstant}}
* {{WolframFunctionsSite |title = Catalan constant: Series representations |urlname = Constants/Catalan/06/01/}}
* {{springer |title = Catalan constant |id = p/c130040 | mode=cs1}}

[[Category:Combinatorics]]
[[Category:Mathematical constants]]