In mathematics, Catalan's constant , is defined by where is the Dirichlet beta function. Its numerical value is approximately 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". 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. 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. 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 The integral is a known special function, called the inverse tangent integral, and was extensively studied by Srinivasa Ramanujan. Relation to other 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. Catalan's constant occurs frequently in relation to the Clausen function, the inverse tangent integral, the inverse sine integral, the Barnes -function, as well