MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

3 deleted 5 characters in body

I'd like to mention that one may prove the irrationality of $\pi$, $\ln 2$, $\zeta(2)$, and $\zeta(3)$ in a relatively uniform way using simple integral representations.

Assume that one wants to show the irrationality of a number $\xi$ which can be presented for every $k\in\mathbb N$ in terms of the moments of some function $f$ $$a_k+b_k\xi=\int_{0}^{1}x^k f(x) dx,$$ where $a_k$, $b_k\in\mathbb Q$. If $\xi$ were rational than the equality might be rewritten as $$\frac{c_n}{d_n}=\int_{0}^{1}P_n(x)f(x)dx,\quad P_n(x)=\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n),\ n\in\mathbb N,$$ where and $c_n$, $d_n$ are some integers. (The choice of the Legendre polynomials $P_n$ allows to perform integrations by parts easily.) Now, if we can show that

$$d_n\left|\int_{0}^{1}\frac{1}{n!}x^n(1-x)^n\frac{d^n}{dx^n}f(x)dx\right|\to 0,$$ this would imply that $c_n\to 0$ which is impossible; so $\xi$ cannot be rational.

The difficult part, of course, is to find the suitable function $f$.

• For $\xi=\pi$ we may take $f(x)=\sin\pi x$ and use the fact that $\int_{0}^{1}x^k\sin(\pi x) dx$ is a polynomial in $\pi$ of degree $k$ divided by $\pi^k$. Assuming $\pi=a/b$ we will get that $$0<|c_n|=\left|a^n\int_{0}^1P_n(x)\sin(\pi x) dx\right|\to 0.$$

• For $\xi=\ln 2$ take $f(x)=1/(1+x)$. If $\ln 2$ were $a/b$, then $$0<|c_n|=\left|bD_n\int_{0}^{1}\frac{P_n(x)}{1+x}dx\right|\to 0$$ (where $D_n={\rm LCM}\{1,2,\dots,n\}$).

• For $\xi=\zeta(2)$ the choice is $$f(x)=\int_{0}^{1}\frac{(1-y)^n}{1-xy}dy,$$ and the assumption $\zeta(2)=a/b$ leads to $$0<|c_n|=\left|D_{n+1}^2\int_{0}^{1}P_n(x)f(x)dx\right|\to 0$$ (where $D_n={\rm LCM}\{1,2,\dots,n\}$).

• Finally, for $\xi=\zeta(3)$ take $$f(x)=\int_{0}^{1}\frac{P_n(y)}{1-xy}\ln xy\ dy.$$ If $\zeta(3)=a/b$, then $$0<|c_n|=\left|D_{n+1}^3\int_{0}^{1}P_n(x)f(x)dx\right|\to 0.$$

The irrationality proofs are scattered contained in the book by J.M. Borwein and P.B. Borwein. There is also a nice summary in the note by D. Huylebrouck (with all four proofs occupying less than five pages).

2 added 407 characters in body

I'd like to mention that one may prove the irrationality of $\pi$, $\ln 2$, $\zeta(2)$, and $\zeta(3)$ in a relatively uniform way using simple integral representations.

Assume that one wants to show the irrationality of a number $\xi$ which can be presented for every $k\in\mathbb N$ in terms of the moments of some function $f$ $$a_k+b_k\xi=\int_{0}^{1}x^k f(x) dx,$$ where $a_k$, $b_k\in\mathbb Q$. If $\xi$ were rational than the equality might be rewritten as $$\frac{c_n}{d_n}=\int_{0}^{1}P_n(x)f(x)dx,\quad P_n(x)=\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n),\ n\in\mathbb N,$$ where and $c_n$, $d_n$ are some integers. (The choice of the Legendre polynomials $P_n$ allows to perform integrations by parts easily.) Now, if we can show that

$$d_n\left|\int_{0}^{1}\frac{1}{n!}x^n(1-x)^n\frac{d^n}{dx^n}f(x)dx\right|\to 0,$$ this would imply that $c_n\to 0$ which is impossible; so $\xi$ cannot be rational.

The difficult part, of course, is to find the suitable function $f$.

• For $\xi=\pi$ we may take $f(x)=\sin\pi x$ and use the fact that $\int_{0}^{1}x^k\sin(\pi x) dx$ is a polynomial in $\pi$ of degree $k$ divided by $\pi^k$. Assuming $\pi=a/b$ we will get that $$0<|c_n|=\left|a^n\int_{0}^1P_n(x)\sin(\pi x) dx\right|\to 0.$$

• For $\xi=\ln 2$ take $f(x)=1/(1+x)$. If $\ln 2$ were $a/b$, then $$0<|c_n|=\left|bD_n\int_{0}^{1}\frac{P_n(x)}{1+x}dx\right|\to 0$$ (where $D_n={\rm LCM}\{1,2,\dots,n\}$).

• For $\xi=\zeta(2)$ the choice is $$f(x)=\int_{0}^{1}\frac{(1-y)^n}{1-xy}dy,$$ and the assumption $\zeta(2)=a/b$ leads to $$0<|c_n|=\left|D_{n+1}^2\int_{0}^{1}P_n(x)f(x)dx\right|\to 0$$ (where $D_n={\rm LCM}\{1,2,\dots,n\}$).

• Finally, for $\xi=\zeta(3)$ take $$f(x)=\int_{0}^{1}\frac{P_n(y)}{1-xy}\ln xy\ dy.$$ If $\zeta(3)=a/b$, then $$0<|c_n|=\left|D_{n+1}^3\int_{0}^{1}P_n(x)f(x)dx\right|\to 0.$$

All

The irrationality proofs are collected scattered in the book by J.M. Borwein and P.B. Borwein. There is also a nice summary in the note by D. Huylebrouck (with all four proofs occupying less than five pages).

1

I'd like to mention that one may prove the irrationality of $\pi$, $\ln 2$, $\zeta(2)$, and $\zeta(3)$ in a relatively uniform way using simple integral representations.

Assume that one wants to show the irrationality of a number $\xi$ which can be presented for every $k\in\mathbb N$ in terms of the moments of some function $f$ $$a_k+b_k\xi=\int_{0}^{1}x^k f(x) dx,$$ where $a_k$, $b_k\in\mathbb Q$. If $\xi$ were rational than the equality might be rewritten as $$\frac{c_n}{d_n}=\int_{0}^{1}P_n(x)f(x)dx,\quad P_n(x)=\frac{1}{n!}\frac{d^n}{dx^n}(x^n(1-x)^n),\ n\in\mathbb N,$$ where and $c_n$, $d_n$ are some integers. (The choice of the Legendre polynomials $P_n$ allows to perform integrations by parts easily.) Now, if we can show that

$$d_n\left|\int_{0}^{1}\frac{1}{n!}x^n(1-x)^n\frac{d^n}{dx^n}f(x)dx\right|\to 0,$$ this would imply that $c_n\to 0$ which is impossible; so $\xi$ cannot be rational.

The difficult part, of course, is to find the suitable function $f$.

• For $\xi=\pi$ we may take $f(x)=\sin\pi x$ and use the fact that $\int_{0}^{1}x^k\sin(\pi x) dx$ is a polynomial in $\pi$ of degree $k$ divided by $\pi^k$. Assuming $\pi=a/b$ we will get that $$0<|c_n|=\left|a^n\int_{0}^1P_n(x)\sin(\pi x) dx\right|\to 0.$$

• For $\xi=\ln 2$ take $f(x)=1/(1+x)$. If $\ln 2$ were $a/b$, then $$0<|c_n|=\left|bD_n\int_{0}^{1}\frac{P_n(x)}{1+x}dx\right|\to 0$$ (where $D_n={\rm LCM}\{1,2,\dots,n\}$).

• For $\xi=\zeta(2)$ the choice is $$f(x)=\int_{0}^{1}\frac{(1-y)^n}{1-xy}dy,$$ and the assumption $\zeta(2)=a/b$ leads to $$0<|c_n|=\left|D_{n+1}^2\int_{0}^{1}P_n(x)f(x)dx\right|\to 0$$ (where $D_n={\rm LCM}\{1,2,\dots,n\}$).

• Finally, for $\xi=\zeta(3)$ take $$f(x)=\int_{0}^{1}\frac{P_n(y)}{1-xy}\ln xy\ dy.$$ If $\zeta(3)=a/b$, then $$0<|c_n|=\left|D_{n+1}^3\int_{0}^{1}P_n(x)f(x)dx\right|\to 0.$$

All proofs are collected in the book by J.M. Borwein and P.B. Borwein.