Apéry's constant ($\zeta(3)$) value
I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + ...$ or there is a easy way to do it?
If there is any sum formula, could you give me the intuition behind this? I really can't just accept formulas, would be great if I could have a explanation.
And about the proof of it being irrational, does someone knows an intuitive link that can explain me this? I only find links that accept things that I don't know. And as I'm asking this only for studying and not for homework, I really wanted to understand as great as possible.
Thanks!
Article $[1]$ by Alfred van der Poorten is a good starting point, the best I know of.
Convergence speed
I mean "brutal force" by summing handly or in a computer, the original formula for $\zeta(3)$.
It is known that the given series converges very slowly. Apéry's rational approximation $a_n/b_n$ below improves the speed of convergence to $\zeta(3)$, while controlling the increase of the size of $b_n$.
Apéry stated the following equality:
$$\zeta (3)\overset{\mathrm{def}}{=}\sum_{n=1}^{\infty }\frac{1}{n^{3}}=\frac{5}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{3}\binom{2n}{n}}.\tag{1}$$
In section 3 van der Poorten shows that
$$\sum_{n=1}^{N}\frac{1}{n^{3}}+\sum_{k=1}^{N}\frac{(-1)^{k-1}}{2k^{3}\binom{N}{k}\binom{N+k}{k}}=\frac{5}{2}\sum_{k=1}^{N }\frac{(-1)^{n-1}}{k^{3}\binom{2k}{k}},\tag{2}$$
from which $(1)$ follows by letting $N$ tend to infinity (see this answer of mine). The formula $(2)$ is not very intuitive though.
The second series in $(1)$ is a fast convergent series, faster by far than the defining series for the Apéry's constant $\zeta(3)$. There are even faster convergent series, obtained by techniques of convergence acceleration.
Proof of the irrationality
As far as the irrationality concerns, Apéry constructed two sequences $(a_n),(b_n)$ $^1$ whose ratio $a_n/b_n\to\zeta(3)$ and
- $2(b_{n}\zeta (3)-a_{n})$ 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. $[2]$.
Intuition
Given that in van der Poorten's words
Those who listened casually, or who were afflicted with being non-Francophone, appeared to hear only a sequence of unlikely assertion.
it is natural to ask: Where did these ideas come from? My tentative explanation is based on the following fact. A few years later after his proof $[3]$, Roger Apéry derived the rational approximation $a_n/b_n$ to $\zeta(3)$ by transforming the defining series for $\zeta(3)$ into a continued fraction and applying iterated transformations to it, which improved the speed of convergence, and obtained the recurrence relations satisfied by $a_{n},b_{n}$ $[4]$.
--
$^1$ The sequences are $$\begin{equation*} a_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2}c_{n,k}, \qquad b_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2},\end{equation*}$$
where
$$\begin{equation*} c_{n,k}=\sum_{m=1}^{n}\frac{1}{m^{3}}+\sum_{m=1}^{k}\frac{\left( -1\right) ^{m-1}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}\quad k\leq n. \end{equation*}$$
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
$[4]$ Apéry, Roger (1981) Interpolations de Fractions Continues et Irrationalité de certaines Constantes, Bull. section des sciences du C.T.H.S., n.º3, p.37-53