论文标题
$ζ（2n）/π^{2n} $合理性的基本证明
An elementary proof of the rationality of $ζ(2n)/π^{2n}$
论文作者
论文摘要
以$ 1735 $ $ euler \ cite {1}证明，对于每个正整数$ k $，系列$ζ（2k）= \ sum _ {\ ell = 1}^{\ infty} \ elfty} \ ell^{ - 2k { - 2k} $收敛到$π^{2k} $的合理倍数。现在已经知道了许多关于这一事实的证明，并且传统上使用非元素技术证明了Euler的发现，例如傅立叶系列或残基的计算\ cite {2}。我们给出一个基本证明，类似于凯奇的\ cite {3}身份$ζ（2）=π^2/6 $的证明，仅对所有值$ζ（2k）$递归扩展。我们的主要公式$$ζ（2K）= - \ dfrac {（ - π^2）^{k}} {4^{2k} -4^{k}}} \ left [\ dfrac {\ dfrac {4^{k} k} k} k} \ sum _ {\ ell = 1}^{k-1}}}（4^{2 \ ell} -4^{\ ell}）\ dfrac {4^{k- \ ell}} {（2k-2 \ ell）！ \ phantom {spa} k = 1,2,3，\ dots $$可以从先前已知的公式\ cite {4}得出。值得注意的是，Apostol \ cite {5}发现了一个与我们类似的证据，但到达了一个不同的公式，将$ζ（2k）$与Bernoulli号码相关联，euler。
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.
