# Why can we not establish the irrationality of Catalan's constant the same way as $\zeta(3)$?

One of the main ingredients in Apéry's proof of the irrationality of $$\zeta(3)$$ is the existence of the fast-converging series:

{\displaystyle {\begin{aligned}\zeta (3)&={\frac {5}{2}}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{{\binom {2k}{k}}k^{3}}}\end{aligned}}}.

Despite numerous attempts, no similar expressions were found for other values of the Riemann $$\zeta$$-function at positive odd integers.

For Catalan's constant, however, we do have such an expression, namely:

$${\displaystyle G={\frac {\pi }{8}}\log \left(2+{\sqrt {3}}\right)+{\frac {3}{8}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}{\binom {2n}{n}}}}.}$$

Why is this not sufficient for applying an Apéry-like method for proving its irrationality?

• Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof. – Dietrich Burde Oct 8 '18 at 12:42