How to prove to be an irrational number? Like $\sqrt{2}$, $\sqrt{3}$, or $\sum\limits_{k=1}^{\infty} \frac{1}{n^2}=\pi^2/6$

Question

As we know $\sqrt{2},\sqrt{3}$ are irrational numbers. And I see some proofs on the net.

So I doubt that how $e,\pi$ or already known irrational numbers are proved to be irrational.

In fact, I got interested in Riemann zeta function $$\zeta(s)=\sum_{n=0}^{\infty} \frac{1}{n^s},$$

we know $\zeta(2)=\pi^2/6$ from Euler, 1737.

One mathematician (sorry to forgot his name) proved $\zeta(3)$ to be also irrational 40 years before.

Can somebody explain how he could do with that? To understand Apéry's theorem, is it very hard?

An question raises that could one real number make up of two different irrationals (for example: $e,\pi$), $e\pi$, or others can be rational?

Answer 1

Some combinations of irrationals can be proved to be irrational in a more or less systematic way. For example, $e^\pi$ is irrational (and in fact transcendental) by Gelfond's theorem (see http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem).

One approach to proving the irrationality of a number is to show that it is approximated "too well" by rational numbers $p/q$ where "too well" is assigned a specific meaning in terms of the denominator $q$. From this point of view, $e$ is irrational because the terms in the familiar power series defining it: $e=\sum 1/n!$ tend to zero rapidly, so that the partial sums give extremely good approximations of $e$.

To get more details, see Spivak's book "Calculus" where he proves irrationality of $\pi$ on page 307, and that $e$ is transcendental on page 409.

Answer 2

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:

1. $2(b_{n}\zeta (3)-a_{n})$ satisfies $\lim\sup \left\vert 2(b_{n}\zeta (3)-a_{n})\right\vert^{1/n}\le(\sqrt{2}-1)^4 $.
2. $b_{n}\in \mathbb{Z},2(\operatorname{lcm}(1,2,\ldots ,n))^{3}a_{n}\in \mathbb{Z}$.
3. $\left\vert b_{n}\zeta (3)-a_{n}\right\vert >0$.

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