An elementary proof of the rationality of $\zeta (2n)/{\pi }^{2n}$

Abstract

In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $\zeta (2k)={\sum }_{\ell =1}^{\infty }{\ell }^{-2k}$ converges to a rational multiple of ${\pi }^{2k}$. 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 $\zeta (2)={\pi }^2/6$, only extended recursively for all values $\zeta (2k)$. Our main formula $$\zeta (2k)=-\frac{(-{\pi }^2{)}^k}{4^{2k}-4^k}\left[\frac{4^{k}k}{(2k)!}+\sum _{\ell =1}^{k-1}(4^{2\ell }-4^{\ell })\frac{4^{k-\ell }}{(2k-2\ell )!}\frac{\zeta (2\ell )}{(-{\pi }^2{)}^{\ell }}\right]k=1,2,3,\dots$$ 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 $\zeta (2k)$ to the Bernoulli numbers, à la Euler.