We establish the triple integral evaluation ∫1∞∫01∫01 dzdydx x(x + y)(x + y + z) = 5 24ζ(3), as well as the equivalent polylogarithmic double sum ∑k=1∞∑ j=k∞(−1)k−1 k2 1 j2j = 13 24ζ(3). This double sum is related to, but less approachable than, similar sums studied by Ramanujan. It is also reminiscent of Euler’s formula ζ(2, 1) = ζ(3), which is the simplest instance of duality of multiple polylogarithms. We review this duality and apply it to derive a companion identity. We also discuss approaches based on computer algebra. All of our approaches ultimately require the introduction of polylogarithms and nontrivial relations between them. It remains an open challenge to relate the triple integral or the double sum to ζ(3) directly.

中文翻译：

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多重 zeta 值的三重积分模拟

我们建立三重积分评估 ∫1∞∫01∫01 dzd是的dX X(X + 是的)(X + 是的 + z) = 5 24ζ(3), 以及等效的多对数双和 ∑ķ=1∞∑ j=ķ∞(-1)ķ-1 ķ2 1 j2j = 13 24ζ(3). 这个双倍和与拉马努金研究的类似总和有关，但不太容易接近。这也让人联想到欧拉公式ζ(2, 1) = ζ(3)，这是多对数对偶的最简单实例。我们回顾了这种二元性并将其应用于派生同伴身份。我们还讨论了基于计算机代数的方法。我们所有的方法最终都需要引入多对数和它们之间的非平凡关系。将三重积分或二重和与ζ(3)直接地。