Catalan's constant

In mathematics, Catalan's constant G is the alternating sum of the reciprocals of the odd square numbers: $${\displaystyle G=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots .}$$ Its numerical value is approximately (sequence A006752 in the OEIS)

G = 0.915965594177219015054603514932384110774...,

and it is also equal to β(2), where β is the Dirichlet beta function.

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".

OpenAI has released a preprint with the title "Catalan's constant is irrational", which the company claims is a proof of irrationality.

However, partial results exist. 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.^{[citation needed]}

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

If E(k) is the complete elliptic integral of the second kind, as a function of the elliptic modulus k, then^{[citation needed]} $${\displaystyle G=-{\frac {1}{2}}+\int _{0}^{1}\mathrm {E} (k)\,dk.}$$

With the gamma function Γ(x + 1) = x!, Catalan's constant is^{[citation needed]} $${\displaystyle {\begin{aligned}G&={\frac {\pi }{4}}\int _{0}^{1}\Gamma \left(1+{\frac {x}{2}}\right)\Gamma \left(1-{\frac {x}{2}}\right)\,dx\\&={\frac {\pi }{2}}\int _{0}^{1/2}\Gamma (1+y)\Gamma (1-y)\,dy.\end{aligned}}}$$

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.^{[citation needed]}

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, π² and Catalan's constant; these are expressible as paths on a graph (see § External links below).

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

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 $${\displaystyle G={\frac {1}{4}}\Phi \left(-1,2,{\tfrac {1}{2}}\right).}$$ This comes directly from the fact that ${\displaystyle \beta (s)=2^{-s}\Phi \left(-1,s,{\tfrac {1}{2}}\right)}$ and ${\displaystyle \beta (2)=G}$, where ${\displaystyle \beta (s)}$ is the Dirichlet beta function

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.

The following continued fraction representation gives (asymptotically) 2.08 new correct decimal places per cycle: $${\displaystyle G={\frac {\frac {13}{2}}{Z_{0}}},\quad Z_{k}=a(k)+{\frac {b(k)}{Z_{k+1}}}}$$ with $${\displaystyle {\begin{aligned}a(k)&=3520k^{6}+5632k^{5}+2064k^{4}-384k^{3}-156k^{2}+16k+7,\\b(k)&=(2k+1)^{4}(2k+2)^{4}(20k^{2}-8k+1)(20k^{2}+72k+65).\end{aligned}}}$$

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

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 0-201-54892-5. 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.{{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)