Catalan's constant

Not to be confused with Catalan number.

Catalan's constant

Rationality	Unknown
Symbol	G
Representations
Decimal	0.9159655941772190150...

In mathematics, Catalan's constantG, 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]}

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. ^[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]}

Properties

Unsolved problem in mathematics:

Is Catalan's constant irrational? If so, is it transcendental?

(more unsolved problems in mathematics)

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.

Series representations

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]

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." ^[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.

Relation to special functions

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

Continued fraction

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.

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. ^[26]

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 [27]
August 2008	10000000000	Shigeru Kondo & Steve Pagliarulo [26]
January 31, 2009	15510000000	Alexander J. Yee & Raymond Chan [28]
April 16, 2009	31026000000	Alexander J. Yee & Raymond Chan [28]
June 7, 2015	200000001100	Robert J. Setti [29]
April 12, 2016	250000000000	Ron Watkins [29]
February 16, 2019	300000000000	Tizian Hanselmann [29]
March 29, 2019	500000000000	Mike A & Ian Cutress [29]
July 16, 2019	600000000100	Seungmin Kim [30] [31]
September 6, 2020	1000000001337	Andrew Sun [32]
March 9, 2022	1200000000100	Seungmin Kim [32]

See also

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

References

1. Papanikolaou, Thomas (March 1997). Catalan's Constant to 1,500,000 Places – via Gutenberg.org.
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..
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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 .
11. 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.
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. hdl:1959.13/803688. 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 .
18. Berndt, B. C. (1985). Ramanujan's Notebook, Part I. Springer Verlag. p. 289. ISBN   978-1-4612-1088-7.
19. Karatsuba, E. A. (1991). "Fast evaluation of transcendental functions". Probl. Inf. Transm. 27 (4): 339–360. MR   1156939. Zbl   0754.65021.
20. 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. ISBN   978-1-4419-3376-8.
21. Alexander Yee (14 May 2019). "Formulas and Algorithms" . Retrieved 5 December 2021.
22. 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.
23. 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.
24. 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.
25. "A014538 - OEIS". oeis.org. Retrieved 2022-10-27.
26. Gourdon, X.; Sebah, P. "Constants and Records of Computation" . Retrieved 11 September 2007.
27. "Shigeru Kondo's website". Archived from the original on 2008-02-11. Retrieved 2008-01-31.
28. "Large Computations" . Retrieved 31 January 2009.
29. "Catalan's constant records using YMP" . Retrieved 14 May 2016.
30. "Catalan's constant records using YMP". Archived from the original on 22 July 2019. Retrieved 22 July 2019.
31. "Catalan's constant world record by Seungmin Kim". 23 July 2019. Retrieved 17 October 2020.
32. "Records set by y-cruncher". www.numberworld.org. Retrieved 2022-02-13.

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 .
• 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].