An elementary proof of the rationality of $ζ(2n)/π^{2n}$

Abstract

In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $ζ(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational multiple of $π^{2k}$. Many demonstrations of this fact are now known, and Euler's discovery is traditionally proven using non-elementary techniques, such as Fourier series or the calculus of residues \cite{2}. We give an elementary proof, similar to Cauchy's \cite{3} proof of the identity $ζ(2) = π^2/6$, only extended recursively for all values $ζ(2k)$. Our main formula $$ζ(2k)=-\dfrac{(-π^2)^{k}}{4^{2k}-4^{k}}\left[\dfrac{4^{k}k}{(2k)!}+{\displaystyle \sum_{\ell=1}^{k-1}}(4^{2\ell}-4^{\ell})\dfrac{4^{k-\ell}}{(2k-2\ell)!}\dfrac{ζ(2\ell)}{(-π^2)^{\ell}}\right] \phantom{spa}k = 1,2,3,\dots$$ may be derived from previously known formulae \cite{4}. Remarkably, Apostol \cite{5} discovered a proof similar to ours, yet arrived at a different formula, relating $ζ(2k)$ to the Bernoulli numbers, à la Euler.