论文标题
多重Zeta值的三个积分类似物
A triple integral analog of a multiple zeta value
论文作者
论文摘要
我们建立三重积分评估\ [\ int_ {1}^{\ infty} \ int_ {0}^{1}^{1} \ int_ {0}^{1}^{1} \ frac {dz \，diz \，dy \，dy \，dx}} = \ frac {5} {24}ζ（3），以及等效的polylogarithmic double sum \ [\ sum_ {k = 1}^{\ infty} \ sum_ { \ frac {1} {j \，2^{j}}} = \ frac {13} {24}ζ（3）。 \]这个双重和与Ramanujan研究的类似总和相关，但易用。它还让人联想到Euler的公式$ζ（2,1）=ζ（3）$，这是多个多聚类的双重性的最简单实例。我们回顾这种二元性并将其应用于获得同伴身份。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各种聚类和它们之间的非平凡关系。直接将三重整数或双款与$ζ（3）$相关联仍然是一个开放的挑战。
We establish the triple integral evaluation \[ \int_{1}^{\infty} \int_{0}^{1} \int_{0}^{1} \frac{dz \, dy \, dx}{x(x+y)(x+y+z)} = \frac{5}{24} ζ(3), \] as well as the equivalent polylogarithmic double sum \[ \sum_{k=1}^{\infty} \sum_{j=k}^{\infty} \frac{(-1)^{k-1}}{k^{2}} \, \frac{1}{j \, 2^{j}} = \frac{13}{24} ζ(3). \] 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.
