Catalan's Constant

Catalan's constant is the alternating sum 1 − 1/9 + 1/25 − 1/49 + ... , the value of the Dirichlet beta function at 2, and it equals 0.915965594177219. Eugène Charles Catalan studied it in 1865, having met it in the evaluation of definite integrals; it turns up in combinatorics as the number of domino tilings and spanning trees on certain lattices, in the volume of the ideal hyperbolic tetrahedron, and repeatedly in two-dimensional lattice models in statistical mechanics.

It sits in the same frustrating category as the Euler–Mascheroni constant. The even values of the related zeta function have closed forms — ζ(2) = π²/6, ζ(4) = π⁴/90 — and so does β(1) = π/4, but β(2) has resisted every attempt at one, and nobody has proved G irrational, let alone transcendental. Billions of digits have been computed using rapidly convergent series due to Ramanujan and others, and every one of them is consistent with irrationality and proves nothing.