How does one prove that $\zeta(3)$ is irrational? How does one prove that $\zeta(3)$ is irrational ?
I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?
 A: The point is that if $\alpha\in\mathbb{Q}^+$ and $C>2$, there is at most a finite number of rational numbers $\frac{p}{q}$ such that:
$$ \left|\alpha-\frac{p}{q}\right|\leq\frac{1}{q^C}. \tag{1}$$
So, if $\left\{\frac{p_n}{q_n}\right\}_{n\in\mathbb{N}}$ is a sequence of rational approximations of $\zeta(3)$ such that $(1)$ holds for every element of the sequence, $\zeta(3)\not\in\mathbb{Q}$. The problem boils down to finding "good" approximations for $\zeta(3)$. This can be done, more or less, by considering a sequence of integrals:
$$ I_n = \int_{-1}^{1} P_n(x)\,f(x)\,dx = \zeta(3)-\frac{p_n}{q_n} \tag{2} $$
where $P_n$ is a Legendre polynomial. To estimate the size of the LHS is not a difficult task, but in order to estimate the size of $q_n$, too, it really useful to know that $q_1,q_2,\ldots$ satisfy some recursive formula. Given the formula, we may compute the magnitude of $q_n$ and hope that $(1)$ holds for some $C>2$ and for an infinite number of $n\in\mathbb{N}$, proving irrationality.
This is far from being an algorithmic approach: there are many well-known cases in which something goes wrong: the Euler-Mascheroni constant $\gamma$, the next interesting value of the $\zeta$ function, $\zeta(5)$, and so on.
Apery was very fond of his proof, and we may figure why: it is almost a miracle.
