It is known to be very difficult to find the arithmetic nature of most real numbers such as the constants that appear in mathematical analysis, e.g. $\zeta (5)=\sum_{n\geq 1}\frac{1}{n^{5}}$.
Roger Apéry proved directly that $\zeta(2)$ is irrational. And he was the first to prove that $\zeta(3)$ is irrational too. He constructed two sequences $(a_n),(b_n)$ $[1]$
$$\begin{equation*}
a_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2}c_{n,k},
\qquad b_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2},\end{equation*}$$
where
$$\begin{equation*}
c_{n,k}=\sum_{m=1}^{n}\frac{1}{m^{3}}+\sum_{m=1}^{k}\frac{\left( -1\right)
^{m-1}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}\quad k\leq n.
\end{equation*}$$
The ratio $a_n/b_n\to\zeta(3)$ and has the following properties:
- $2(b_{n}\zeta (3)-a_{n})$ satisfies $\lim\sup \left\vert 2(b_{n}\zeta (3)-a_{n})\right\vert^{1/n}\le(\sqrt{2}-1)^4 $.
- $b_{n}\in \mathbb{Z},2(\operatorname{lcm}(1,2,\ldots ,n))^{3}a_{n}\in
\mathbb{Z}$.
- $\left\vert b_{n}\zeta (3)-a_{n}\right\vert >0$.
This is enough to prove the irrationality of $\zeta (3)$ by contradiction. $[2]$.
There is The Tricki entry To prove that a number is irrational, show that it is almost rational that gives two examples and explains the principle of some proofs of the irrationality of numbers.
References.
$[1]$ Poorten, Alf., A Proof that Euler Missed…, Apéry’s proof of the irrationality of $\zeta(3)$. An informal report, Math. Intelligencer 1, nº 4, 1978/79, pp. 195-203.
$[2]$ Fischler, Stéfane, Irrationalité de valeurs de zêta (d’ après Apéry, Rivoal, …), Séminaire Bourbaki 2002-2003, exposé nº 910 (nov. 2002), Astérisque 294 (2004), 27-62
$[3]$ Apéry, Roger (1979), Irrationalité de $\zeta2$ et $\zeta3$, Astérisque 61: 11–13
