Number, approximately 0.916
In mathematics, Catalan's constant G, is the alternating sum of the reciprocals of the odd square numbers, being defined by:
 formula_1
where β is the Dirichlet beta function. Its numerical value is approximately (sequence in the OEIS)
 "G" = …
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 formula_2 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.
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Unsolved problem in mathematics:
Is Catalan's constant irrational? If so, is it transcendental?
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"(2"n") 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(2"n+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:formula_3formula_4
The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:
formula_5
and
formula_6
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, formula_7.
Other quickly converging series, due to Guillera and Pilehrood and employed by the y-cruncher software, include:
formula_8
formula_9
formula_10
All of these series have time complexity formula_11.
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:
formula_12
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
formula_13
If E("k") is the complete elliptic integral of the second kind, as a function of the elliptic modulus "k", then
formula_14
With the gamma function Γ("x" + 1) = "x"!
formula_15
The integral
formula_16
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:
formula_17
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):
formula_18
If one defines the Lerch transcendent Φ("z","s","α") by
formula_19
then
formula_20
Continued fraction.
G can be expressed in the following form:
formula_21
The simple continued fraction is given by:
formula_22
This continued fraction would have infinite terms if and only if formula_23 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.
References.
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