标题： 费曼积分，L级数和克卢斯特曼矩 显示英文标题

标题： Feynman integrals, L-series and Kloosterman moments

摘要： 这项工作位于三个学科的交汇点：量子场论、代数几何和数论，在从业者之间的对话中揭示了丰富的结构。 它包含一个定理和7个猜想，通过3个优化算法进行了深入测试，涉及费曼积分与由有限域中克洛斯特曼和的矩确定的数据在素数上的乘积定义的L级数之间的关系。 有一个扩展的引言，供可能不熟悉这三个学科的读者阅读。 显著的新结果包括通过费曼积分行列式对权为3、4和6的模形式的非临界L级数的猜想评估，对权5问题在临界整数处的评估，以及任意大小行列式的公式，测试到了30圈。 证明了克洛斯特曼矩的函数方程决定了L级数结构的大部分，但不是全部。 特别是，对于包含奇数个贝塞尔函数的问题，它遗漏了一个关键特征，该特征在本工作中通过新颖且经过严格测试的猜想得以捕捉。 对于9贝塞尔问题，这些猜想导致了在素数上的数据惊人压缩。 显示英文摘要

摘要： This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and L-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical L-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the L-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes.