量子场论中双对数积分的Borwein-Broadhurst猜想的证明
Proof of the Borwein-Broadhurst conjecture for a dilogarithmic integral arising in quantum field theory
摘要：
Borwein和Broadhurst在1998年利用实验数学技术，发现了许多双曲三维流形，它们的体积与各种狄利克雷L级数$\textup{L}_{d}(s)$的值有合理的关系。特别是，在双曲空间中理想四面体的最简单情况下，他们证明了表示体积的双对数积分等于$\textup{L}_{-7}(2)$。在这里，我们提供了这个猜想的正式证明，最近在尖端计算实验中得到了数值验证（至少19，995位，在1024个处理器上使用45分钟）。证明基本上依赖于结果Zagier的公式的价值Dedekind zeta函数$ζ_{\mathbb{K}}(2)$的任意领域$\mathbb{K}$。
英文摘要：
Borwein and Broadhurst, using experimental-mathematics techniques, in 1998 identified numerous hyperbolic 3-manifolds whose volumes are rationally related to values of various Dirichlet L series $\textup{L}_{d}(s)$. In particular, in the simplest case of an ideal tetrahedron in hyperbolic space, they conjectured that a dilogarithmic integral representing the volume equals to $\textup{L}_{-7}(2)$. Here we have provided a formal proof of this conjecture which has been recently numerically verified (to at least 19,995 digits, using 45 minutes on 1024 processors) in cutting-edge computing experiments. The proof essentially relies on the results of Zagier on the formula for the value of Dedekind zeta function $ζ_{\mathbb{K}}(2)$ for an arbitrary field $\mathbb{K}$.
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