# Mock Theta Functions

@article{Zwegers2008MockTF, title={Mock Theta Functions}, author={Sander P. Zwegers}, journal={arXiv: Number Theory}, year={2008} }

The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death. In this dissertation, I consider several of the examples that Ramanujan gave of mock theta functions, and relate them to real-analytic modular forms of weight 1/2.
In Chapter 1, I consider a sum, which was also studied by Lerch. This Lerch sum transforms almost as a Jacobi form under substitutions in (upsilon, nu, tau ). I show…

## 531 Citations

A Survey of Classical Mock Theta Functions

- Mathematics
- 2012

In his last letter to Hardy, Ramanujan defined 17 functions M(q), | q | < 1, which he called mock θ-functions. He observed that as q radially approaches any root of unity ζ at which M(q) has an…

Partial theta functions and mock modular forms as q-hypergeometric series

- Mathematics
- 2011

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious…

Jacobi's triple product, mock theta functions, and the $q$-bracket

- Mathematics
- 2016

In Ramanujan’s final letter to Hardy, he wrote of a strange new class of infinite series he called “mock theta functions”. It turns out all of Ramanujan’s mock theta functions are essentially…

Partial New Mock Theta Functions

- Mathematics
- 2012

By using a simple identity of mine, I have proved sixty identities connecting the mock theta functions of Ramanujan with mock theta functions recently generated by Andrews and Bringmann et al, by…

Note added in proof

- MathematicsThe Botanical Review
- 2008

The importance of the transformation laws of Gordon and McIntosh plays a central role in clarifying how several infinite families of mock theta functions sit within the infinite dimensional spaces of Maass forms are clarified.

RAMANUJAN’S RADIAL LIMITS

- Mathematics
- 2013

Ramanujan’s famous deathbed letter to G. H. Hardy concerns the asymptotic properties of modular forms and his so-called mock theta functions. For his mock theta function f(q), he asserts, as q…

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

- MathematicsPhilosophical Transactions of the Royal Society A
- 2019

The author 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.

Weierstrass mock modular forms and elliptic curves

- Mathematics
- 2014

AbstractMock modular forms, which give the theoretical framework for Ramanujan’s enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular…

Contemporary Mathematics Ramanujan ’ s radial limits

- Mathematics
- 2013

Ramanujan’s famous deathbed letter to G. H. Hardy concerns the asymptotic properties of modular forms and his so-called mock theta functions. For his mock theta function f(q), he asserts, as q…

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In a recent paper [H], we proved the "Mock Theta Conjectures". These are identities, stated by Ramanujan in his "lost notebook" JR2, pp. 19-20], involving two of the 5th order mock 0-functions. In…

Tenth order mock theta functions in Ramanujan’s lost notebook (IV)

- Mathematics
- 2001

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved…

Tenth order mock theta functions in Ramanujan's Lost Notebook

- Mathematics
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Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved…

Tenth Order Mock Theta Functions in Ramanujan's Lost Notebook II

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In S. Ramanujan's last letter to G. H. Hardy [B4], Ramanujan proclaimed, ``I discovered very interesting functions recently which I call `Mock' -functions.'' He then provided a long list of ``third…

Some Eighth Order Mock Theta Functions

- Mathematics
- 2000

A method is developed for obtaining Ramanujan's mock theta functions from ordinary theta functions by performing certain operations on their q‐series expansions. The method is then used to construct…

MOCK θ-FUNCTIONS AND REAL ANALYTIC MODULAR

- Mathematics
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In this paper we examine three examples of Ramanujan’s third order mock θ-functions and relate them to Rogers’ false θ-series and to a real-analytic modular form of weight 1/2.