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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?

Tito Piezas III
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Klangen
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    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 08 '18 at 12:42
  • Even if such expression were sufficient to prove irrationality, it would only prove it for the series. Then the question of the irrationality of the sum of the series and $\frac{\pi}{8}\log(2+\sqrt{3})$ would still be another question I think – Giafazio Jan 30 '23 at 14:37
  • @Giafazio If could be that the right hand term is a linear combination of one or more of the constants in the left hand term though – Klangen Jan 30 '23 at 14:50

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Maybe the central binomial series I've discovered in arXiv paper https://arxiv.org/abs/1207.3139, namely $$G = -\,\frac{1}{2} \, \sum_{n=1}^\infty{(-1)^n \, \frac{(3n-1)\,8^n}{(2n+1)^3 \, \binom{2n}{n}^3}}$$, can be useful for you. There in that paper I've shown that the convergence rate of my series is better than that you mentioned above! Regards, Fabio

Dr. Fabio M. S. Lima
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