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

TL;DR

This paper presents an elementary recursive proof that the values of the Riemann zeta function at even integers are rational multiples of , avoiding complex analysis techniques.

Contribution

It provides a new elementary proof of the rationality of () for all positive integers k, extending Cauchy's approach to all even zeta values.

Findings

Derived a recursive formula for () involving elementary functions.

Confirmed the rationality of () without advanced techniques.

Connected the proof to classical results and previous formulas.

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 $$\zeta(2k)=-\dfrac{(-\pi^2)^{k}}{4^{2k}-4^{k}}\left[\dfrac{4^{k}k}{(2k)!}+{\displaystyle \sum_{\ell=1}^{k-1}}(4^{2\ell}-4^{\ell})\dfrac{4^{k-\ell}}{(2k-2\ell)!}\dfrac{\zeta(2\ell)}{(-\pi^2)^{\ell}}\right] \phantom{spa}k = 1,2,3,\dots$$ 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 $\zeta(2k)$ to the Bernoulli numbers, \`a la Euler.