Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
Proc Natl Acad Sci U S A. 2013 May 7;110(19):7592-4. doi: 10.1073/pnas.1301046110. Epub 2013 Apr 26.
In his famous "deathbed" letter, Ramanujan "defined" the notion of a mock theta function and offered some examples of functions he believed satisfied his definition. Very recently, Griffin et al. established for the first time that Ramanujan's mock theta functions actually satisfy his own definition. On the other hand, Zwegers' 2002 doctoral thesis [Zwegers S (2002) Mock theta functions. PhD thesis (Univ Utrecht, Utrecht, The Netherlands)] showed that all of Ramanujan's examples are holomorphic parts of harmonic Maass forms. This has led to an alternate definition of a mock theta function. This paper shows that Ramanujan's definition of mock theta function is not equivalent to the modern definition.
在他著名的“临终”信中，拉马努金“定义”了拟theta 函数的概念，并提供了一些他认为满足其定义的函数示例。最近，Griffin 等人首次证实，拉马努金的拟theta 函数实际上满足他自己的定义。另一方面，Zwegers 于 2002 年的博士论文[Zwegers S（2002）Mock theta functions. PhD thesis (Univ Utrecht, Utrecht, The Netherlands)]表明，拉马努金的所有例子都是调和 Maass 形式的全纯部分。这导致了对拟theta 函数的另一种定义。本文表明，拉马努金拟theta 函数的定义与现代定义并不等价。
