In mathematics, Catalan's constant G is the alternating sum of the reciprocals of the odd square numbers:
G = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 1 1 2 − 1 3 2 + 1 5 2 − 1 7 2 + 1 9 2 − ⋯ . {\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 n 2 + 1 {\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". 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.
Series representations Catalan's constant appears in the evaluation of several rational series including: π 2 16 + G 2 = ∑ n = 0 ∞ 1 ( 4 n + 1 ) 2 . {\displaystyle {\frac {\pi ^{2}}{16}}+{\frac {G}{2}}=\sum _{n=0}^{\infty }{\frac {1}{(4n+1)^{2}}}.}
π 2 16 − G 2 = ∑ n = 0 ∞ 1 ( 4 n + 3 ) 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:
G = 3 ∑ n = 0 ∞ 1 2 4 n ( − 1 2 ( 8 n + 2 ) 2 + 1 2 2 ( 8 n + 3 ) 2 − 1 2 3 ( 8 n + 5 ) 2 + 1 2 3 ( 8 n + 6 ) 2 − 1 2 4 ( 8 n + 7 ) 2 + 1 2 ( 8 n + 1 ) 2 ) − 2 ∑ n = 0 ∞ 1 2 12 n ( 1 2 4 ( 8 n + 2 ) 2 + 1 2 6 ( 8 n + 3 ) 2 − 1 2 9 ( 8 n + 5 ) 2 − 1 2 10 ( 8 n + 6 ) 2 − 1 2 12 ( 8 n + 7 ) 2 + 1 2 3 ( 8 n + 1 ) 2 ) {\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
G = π 8 log ( 2 + 3 ) + 3 8 ∑ n = 0 ∞ 1 ( 2 n + 1 ) 2 ( 2 n n ) . {\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, ζ ( 3 ) {\displaystyle \zeta (3)} . Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:
G = 1 2 ∑ k = 0 ∞ ( − 8 ) k ( 3 k + 2 ) ( 2 k + 1 ) 3 ( 2 k k ) 3 {\displaystyle G={\frac {1}{2}}\sum _{k=0}^{\infty }{\frac {(-8)^{k}(3k+2)}{(2k+1)^{3}{\binom {2k}{k}}^{3}}}}
G = 1 64 ∑ k = 1 ∞ 256 k ( 580 k 2 − 184 k + 15 ) k 3 ( 2 k − 1 ) ( 6 k 3 k ) ( 6 k 4 k ) ( 4 k 2 k ) {\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}}}}}
G = − 1 1024 ∑ k = 1 ∞ ( − 4096 ) k ( 45136 k 4 − 57184 k 3 + 21240 k 2 − 3160 k + 165 ) k 3 ( 2 k − 1 ) 3 ( ( 2 k ) ! 6 ( 3 k ) ! 3 k ! 3 ( 6 k ) ! 3 ) {\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 O ( n log ( n ) 3 ) {\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:
G = − 1 π i ∫ 0 π / 2 ln ln tan x ln tan x d x , G = ∬ [ 0 , 1 ] 2 1 1 + x 2 y 2 d x d y , G = ∫ 0 1 ∫ 0 1 − x 1 1 − x 2 − y 2 d y d x , G = ∫ 1 ∞ ln t 1 + t 2 d t , G = − ∫ 0 1 ln t 1 + t 2 d t , G = 1 2 ∫ 0 π / 2 t sin t d t , G = ∫ 0 π / 4 ln cot t d t , G = 1 2 ∫ 0 π / 2 ln ( sec t + tan t ) d t , G = ∫ 0 1 arccos t 1 + t 2 d t , G = ∫ 0 1 arcsinh t 1 − t 2 d t , G = 1 2 ∫ 0 ∞ arctan t t 1 + t 2 d t , G = 1 2 ∫ 0 1 arctanh t 1 − t 2 d t , G = ∫ 0 ∞ arccot e t d t , G = 1 4 ∫ 0 π 2 / 4 csc t d t , G = 1 16 ( π 2 + 4 ∫ 1 ∞ arccsc 2 t d t ) , G = 1 2 ∫ 0 ∞ t cosh t d t , G = π 2 ∫ 1 ∞ ( t 4 − 6 t 2 + 1 ) ln ln t ( 1 + t 2 ) 3 d t , G = 1 2 ∫ 0 ∞ arcsin ( sin t ) t d t , G = 1 + lim α → 1 −