Asymptotics and Ramanujan's mock theta functions: then and now

Abstract

This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms—Dyson called this ‘a challenge for the future’ at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later in 2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann–Rolen, Choi–Lim–Rhoades and Griffin–Ono–Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.