Catalan's constant

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^[1] 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.^[2][3]

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.^[4] It is 1/8 of the volume of the complement of the Borromean rings.^[5]

In combinatorics and statistical mechanics, it arises in connection with counting domino tilings,^[6] spanning trees,^[7] and Hamiltonian cycles of grid graphs.^[8]

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.^[9]

Catalan's constant also appears in the calculation of the mass distribution of spiral galaxies.^[10][11]

It is not known whether G is irrational, let alone transcendental.^[12] G has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven".^[13]

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.^[14] 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.^[15] 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.

Catalan's constant appears in the evaluation of several rational series including:^[16]$${\displaystyle {\frac {\pi ^{2}}{16}}+{\frac {G}{2}}=\sum _{n=0}^{\infty }{\frac {1}{(4n+1)^{2}}}.}$$$${\displaystyle {\frac {\pi ^{2}}{16}}-{\frac {G}{2}}=\sum _{n=0}^{\infty }{\frac {1}{(4n+3)^{2}}}.}$$ 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,^[17] and Ramanujan, for the second formula.^[18] The algorithms for fast evaluation of the Catalan constant were constructed by E. Karatsuba.^[19][20] Using these series, calculating Catalan's constant is now about as fast as calculating Apéry's constant, ${\displaystyle \zeta (3)}$.^[21]

Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:^[21]

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

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

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

All of these series have time complexity ${\displaystyle O(n\log(n)^{3})}$.^[21]

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."^[22] 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.^[23]

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.

G appears in values of the second polygamma function, also called the trigamma function, at fractional arguments:^[16]

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

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

G can be expressed in the following form:^[24]

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

The simple continued fraction is given by:^[25]

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

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

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.^[26]

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

• 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.

• 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 [1994].{{citation}}: CS1 maint: overridden setting (link)