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In mathematics, Catalan's constant G, is the alternating sum of the reciprocals of the odd square numbers, being 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 is approximately (sequence A006752 in the OEIS)

G = 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.

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 ${\displaystyle n^{2}+1}$ 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.

Catalan's constant also appears in the calculation of the mass distribution of spiral galaxies.

Properties

It is not known whether G is irrational, let alone transcendental. G 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:$${\displaystyle {\frac {\pi ^{2}}{16}}+{\frac {G}{2}}=\sum _{n=0}^{\infty }{\frac {1}{(4n+1)^{2}}}.}$$$${\displaystyle {\frac {\pi ^{2}}{16}}-{\frac {G}{2}}=\sum _{n=0}^{\infty }{\frac {1}{(4n+3)^{2}}}.}$$ 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, 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, ${\displaystyle \zeta (3)}$.

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

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

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

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

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

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: $${\displaystyle {\begin{aligned}G&=-{\frac {1}{\pi i}}\int _{0}^{\frac {\pi }{2}}\ln \ln \tan x\ln \tan x\,dx\\G&=\iint _{^{2}}\!{\frac {1}{1+x^{2}y^{2}}}\,dx\,dy\\G&=\int _{0}^{1}\int _{0}^{1-x}{\frac {1}{1-x^{2}-y^{2}}}\,dy\,dx\\G&=\int _{1}^{\infty }{\frac {\ln t}{1+t^{2}}}\,dt\\G&=-\int _{0}^{1}{\frac {\ln t}{1+t^{2}}}\,dt\\G&={\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}{\frac {t}{\sin t}}\,dt\\G&=\int _{0}^{\frac {\pi }{4}}\ln \cot t\,dt\\G&={\frac {1}{2}}\int _{0}^{\frac {\pi }{2}}\ln \left(\sec t+\tan t\right)\,dt\\G&=\int _{0}^{1}{\frac {\arccos t}{\sqrt {1+t^{2}}}}\,dt\\G&=\int _{0}^{1}{\frac {\operatorname {arcsinh} t}{\sqrt {1-t^{2}}}}\,dt\\G&={\frac {1}{2}}\int _{0}^{\infty }{\frac {\operatorname {arctan} t}{t{\sqrt {1+t^{2}}}}}\,dt\\G&={\frac {1}{2}}\int _{0}^{1}{\frac {\operatorname {arctanh} t}{\sqrt {1-t^{2}}}}\,dt\\G&=\int _{0}^{\infty }\operatorname {arccot} e^{t}\,dt\\G&={\frac {1}{4}}\int _{0}^{{\pi ^{2}}/{4}}\csc {\sqrt {t}}\,dt\\G&={\frac {1}{16}}\left(\pi ^{2}+4\int _{1}^{\infty }\operatorname {arccsc} ^{2}t\,dt\right)\\G&={\frac {1}{2}}\int _{0}^{\infty }{\frac {t}{\cosh t}}\,dt\\G&={\frac {\pi }{2}}\int _{1}^{\infty }{\frac {\left(t^{4}-6t^{2}+1\right)\ln \ln t}{\left(1+t^{2}\right)^{3}}}\,dt\\G&={\frac {1}{2}}\int _{0}^{\infty }{\frac {\arcsin \left(\sin t\right)}{t}}\,dt\\G&=1+\lim _{\alpha \to {1^{-}}}\!\left\{\int _{0}^{\alpha }\!{\frac {\left(1+6t^{2}+t^{4}\right)\arctan {t}}{t\left(1-t^{2}\right)^{2}}}\,dt+2\operatorname {artanh} {\alpha }-{\frac {\pi \alpha }{1-\alpha ^{2}}}\right\}\\G&=1-{\frac {1}{8}}\iint _{\mathbb {R} ^{2}}\!\!{\frac {x\sin \left(2xy/\pi \right)}{\,\left(x^{2}+\pi ^{2}\right)\cosh x\sinh y\,}}\,dx\,dy\\G&=\int _{0}^{\infty }\int _{0}^{\infty }{\frac {{\sqrt{x}}\left({\sqrt {x}}{\sqrt {y}}-1\right)}{(x+1)^{2}{\sqrt{y}}(y+1)^{2}\log(xy)}}dxdy\end{aligned}}}$$

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

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 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, π 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) by $${\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}},}$$ then $${\displaystyle G={\tfrac {1}{4}}\Phi \left(-1,2,{\tfrac {1}{2}}\right).}$$

Continued fraction

G can be expressed in the following form:

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

The simple continued fraction is given by:

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

This continued fraction would have infinite terms if and only if ${\displaystyle G}$ is irrational, which is still unresolved.

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.

See also

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

References

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2. Goldstein, Catherine (2015). "The mathematical achievements of Eugène Catalan". Bulletin de la Société Royale des Sciences de Liège. 84: 74–92. MR 3498215.
3. 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.
4. 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..
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12. 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..
13. 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.
14. Rivoal, T.; Zudilin, W. (2003-08-01). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen. 326 (4): 705–721. doi:10.1007/s00208-003-0420-2. ISSN 1432-1807.
15. Zudilin, Wadim (2018-04-26). "Arithmetic of Catalan's constant and its relatives". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 89: 45–53. arXiv:1804.09922. doi:10.1007/s12188-019-00203-w.
16. ^ Weisstein, Eric W. "Catalan's Constant". mathworld.wolfram.com. Retrieved 2024-10-02.
17. Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)". arXiv:math.CA/9803067.
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Further reading

• Adamchik, Victor (2002). "A certain series associated with Catalan's constant". Zeitschrift für Analysis und ihre Anwendungen. 21 (3): 1–10. doi:10.4171/ZAA/1110. MR 1929434.
• Fee, Gregory J. (1990). "Computation of Catalan's Constant Using Ramanujan's Formula". In Watanabe, Shunro; Nagata, Morio (eds.). Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC '90, Tokyo, Japan, August 20-24, 1990. ACM. pp. 157–160. doi:10.1145/96877.96917. ISBN 0201548925. S2CID 1949187.
• Bradley, David M. (1999). "A class of series acceleration formulae for Catalan's constant". The Ramanujan Journal. 3 (2): 159–173. arXiv:0706.0356. Bibcode:2007arXiv0706.0356B. doi:10.1023/A:1006945407723. MR 1703281. S2CID 5111792.

External links

• Adamchik, Victor. "33 representations for Catalan's constant". Archived from the original on 2016-08-07. Retrieved 14 July 2005.
• Plouffe, Simon (1993). "A few identities (III) with Catalan". Archived from the original on 2019-06-26. Retrieved 29 July 2005. (Provides over one hundred different identities).
• Plouffe, Simon (1999). "A few identities with Catalan constant and Pi^2". Archived from the original on 2019-06-26. Retrieved 29 July 2005. (Provides a graphical interpretation of the relations)
• Fee, Greg (1996). Catalan's Constant (Ramanujan's Formula). (Provides the first 300,000 digits of Catalan's constant)
• Bradley, David M. (2001). Representations of Catalan's constant. CiteSeerX 10.1.1.26.1879.{{citation}}: CS1 maint: overridden setting (link)
• Johansson, Fredrik. "0.915965594177219015054603514932". Ordner, a catalog of real numbers in Fungrim. Retrieved 21 April 2021.
• "Catalan's Constant". YouTube. Let's Learn, Nemo!. 10 August 2020. Retrieved 6 April 2021.
• Weisstein, Eric W. "Catalan's Constant". MathWorld.
• "Catalan constant: Series representations". Wolfram Functions Site.
• "Catalan constant". Encyclopedia of Mathematics. EMS Press. 2001 .{{citation}}: CS1 maint: overridden setting (link)

• Combinatorics
• Mathematical constants