On Ramanujan's definition of mock theta function

Abstract

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.