Ueber Eigenwerte，Integrale und pi^2/6：Die Idee der Spurformel（论特征值、积分和pi^2/6：迹公式的思想）
Ueber Eigenwerte, Integrale und pi^2/6: Die Idee der Spurformel (On eigenvalues, integrals and pi^2/6: The idea of the trace formula)
摘要：
这是一篇简短的文章，来自奥尔登堡大学二年级学生的演讲。谈话的目的是显示什么美丽和意想不到的结果可能会获得，如果一个人玩大胆的类比的方式，通常不这样做，在本科教育（不幸的是）：我们开始从一个事实，即总和的对角条目的对称矩阵等于其特征值的总和。然后，我们猜测一个类似的公式，其中矩阵被替换为两个实变量的函数和总和被替换为积分在一个系统的方式。我们证明了这确实是一个有价值的过程：在一个特殊情况下，它得出正整数的平方倒数之和是π ^2/6。最后，给出了猜测公式的证明大纲，并解释了进一步的应用，例如台球和鼓的频率之间的联系。
英文摘要：
This is an expository article that results from a talk given to second year students at Oldenburg university. The aim of the talk was to show what beautiful and unexpected results may be obtained if one plays with daring analogies in a way that is usually not done in undergraduate education (unfortunately): We start from the fact that the sum of diagonal entries of a symmetric matrix equals the sum of its eigenvalues. We then guess an analogous formula where the matrix is replaced by a function of two real variables and sums are replaced by integrals in a systematic way. We show that this is indeed a worthwhile process: In a special case it yields that the sum of inverse squares of the positive integers is pi^2/6. Finally, an outline of the proof of the guessed formula is given, and further applications, for example to the connection between billiards and the frequencies of a drum, are explained.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
