Riemann Zeta Function at Even Integers/Lemma/Proof 1

Lemma

Let $x \in \R$ be such that $\size x < 1$.

Then:

$\ds \pi x \cot {\pi x} = 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}$

where $\zeta$ denotes the Riemann zeta function.

$\ds \frac {\sin \pi x} {\pi x}$

$=$

$\ds \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }$

$\ds \leadsto \ \ $

$\ds \map \ln {\frac {\sin \pi x} {\pi x} }$

$=$

$\ds \ln \prod_{k \mathop = 1}^\infty \paren {1 - \frac {x^2} {k^2} }$

$\ds \leadsto \ \ $

$\ds \map \ln {\sin {\pi x} }$

$=$

$\ds \map \ln {\pi x} + \sum_{k \mathop = 1}^\infty \map \ln {1 - \frac {x^2} {k^2} }$

$\ds \leadsto \ \ $

$\ds \pi \frac {\cos {\pi x} } {\sin {\pi x} }$

$=$

$\ds \frac 1 x + \sum_{k \mathop = 1}^\infty \frac 1 {\paren {1 - \frac {x^2} {k^2} } } \paren {-\frac {2 x} {k^2} }$

$\ds \leadsto \ \ $

$\ds \pi x \cot {\pi x}$

$=$

$\ds 1 + \sum_{k \mathop = 1}^\infty \frac 1 {\paren {1 - \frac {x^2} {k^2} } } \paren {-\frac {2 x^2} {k^2} }$

$\ds $

$=$

$\ds 1 + \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 0}^\infty \paren {\frac {x^2} {k^2} }^n \paren {-\frac {2 x^2} {k^2} }$

$\ds $

$=$

$\ds 1 - 2 \sum_{k \mathop = 1}^\infty \sum_{n \mathop = 1}^\infty \paren {\frac {x^2} {k^2} }^n$

$\ds $

$=$

$\ds 1 - 2 \sum_{n \mathop = 1}^\infty \paren {\sum_{k \mathop = 1}^\infty \frac 1 {k^{2 n} } } x^{2 n}$

interchanging order of summation is valid by Tonelli's Theorem

$\ds $

$=$

$\ds 1 - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} x^{2 n}$

$\blacksquare$