Catalan's constant

In mathematics, Catalan's constant is the alternating sum of the reciprocals of the odd square numbers:

Its numerical value is approximately , and it is also equal to , where 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 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 is irrational, let alone transcendental.. 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

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: The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

and

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, .

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

All of these series have time complexity .

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:

where the last three formulas are related to Malmsten's integrals.

If is the complete elliptic integral of the first kind, as a function of the elliptic modulus , then

If is the complete elliptic integral of the second kind, as a function of the elliptic modulus , then

With the gamma function , Catalan's constant is

The integral

is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan.

Relation to special functions appears in values of the second polygamma function, also called the trigamma function, at fractional arguments:

Simon Plouffe gives an infinite collection of identities between the trigamma function, 2 and Catalan's constant; these are expressible as paths on a graph (see below).

Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes -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 -function, the following expression is obtained (see Clausen function for more):

If one defines the Lerch transcendent by

then

This comes directly from the fact that and , where is the Dirichlet beta function

Continued fraction can be expressed in the following form:

The simple continued fraction is given by

This continued fraction would have infinite terms if and only if is irrational, which is still unresolved.

The following continued fraction representation gives (asymptotically) 2.08 new correct decimal places per cycle:

with

Known digits The number of known digits of Catalan's constant has increased dramatically during the last decades. This is due both to the increase of performance of computers as well as to algorithmic improvements.

+ Number of known decimal digits of Catalan's constant 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 Greg J. Fee 1996 Greg J. Fee August 14, 1996 Greg J. Fee & Simon Plouffe September 29, 1996 Thomas Papanikolaou 1996 Thomas Papanikolaou 1997 Patrick Demichel January 4, 1998 Xavier Gourdon 2001 Xavier Gourdon & Pascal Sebah 2002 Xavier Gourdon & Pascal Sebah October 2006 Shigeru Kondo & Steve Pagliarulo August 2008 Shigeru Kondo & Steve Pagliarulo January 31, 2009 Alexander J. Yee & Raymond Chan April 16, 2009 Alexander J. Yee & Raymond Chan June 7, 2015 Robert J. Setti April 12, 2016 Ron Watkins February 16, 2019 Tizian Hanselmann March 29, 2019 Mike A & Ian Cutress July 16, 2019 Seungmin Kim September 6, 2020 Andrew Sun March 9, 2022 Seungmin Kim

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

References

Catalan's constant

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