标题： 拉马努金、兰道和卡西米尔，发散级数：物理学家的观点 显示英文标题

标题： Ramanujan, Landau and Casimir, divergent series: a physicist point of view

摘要： 一个广为人知的悖论练习是证明正整数的无穷和等于 -1/12，有时称为拉马努金和。 在这里，我们提出一种定性的方法，类似于物理学家的方法，来展示为何 -1/12 这个值是有意义的，并且事实上出现在涉及这种求和的某些物理量中。 通过两个物理例子，分别来自凝聚态物质——朗道反磁性——和量子电动力学——卡西米尔效应——来说明这个奇怪的求和，我们介绍了一种系统的方法来从无穷大中提取出拉马努金项。 显示英文摘要

摘要： It is a popular paradoxical exercise to show that the infinite sum of positive integer numbers is equal to -1/12, sometimes called the Ramanujan sum. Here we propose a qualitative approach, much like that of a physicist, to show how the value -1/12 can make sense and, in fact, appears in certain physical quantities where this type of summation is involved. At the light of two physical examples, taken respectively from condensed matter -- the Landau diamagnetism -- and quantum electrodynamics -- the Casimir effect -- that illustrate this strange sum, we present a systematic way to extract this Ramanujan term from the infinity.