Mock Theta Functions

By Sander Zwegers. 2008

183 citations recorded by the source corpus.

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 that the transformation behaviour becomes that of a Jacobi form if we add a (relatively simple) correction term. This correction term is real-analytic in (upsilon, nu, tau) but not holomorphic. For special values of (upsilon, nu), we could call the Lerch sum (considered as a function of tau ) a mock theta function, although these examples were not considered by Ramanujan. In Chapter 2, I consider theta functions for indefinite quadratic forms. These indefinite theta functions are modified versions of theta functions introduced by Gottsche and Zagier. I find elliptic and modular transformation properties of these functions, similar to the properties of theta functions associated to positive definite quadratic forms. In the case of positive definite quadratic forms, the theta functions are holomorphic. The theta functions in the chapter are not holomorphic. By taking special values of certain parameters, we get most of the examples of mock theta functions given by Ramanujan. Andrews gave most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients. In Chapter 4, I use the results from Chapter 2 to get the modular transformation properties of the seventh-order mock v-functions and most of the fifth-order functions. The final result is that we can write each of these mock theta-functions as the sum of two functions ? and G, where: - ? is a real-analytic modular form of weight 1/2 and is an eigenfunction of the appropriate Casimir operator with eigenvalue 3/16 (this is also the eigenvalue of holomorphic modular forms of this weight); and - G is a theta series associated to a negative definite unary quadratic form. Moreover G is bounded as ? tends vertically to any rational limit. Many of the results of Chapter 4 could be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter 2. This means that I have actually given three approaches to proving modularity properties of the mock theta -functions.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/107a8dae22effe7a429a7c8ecc969df12d12e659

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Citations in the corpus

Eichler–Shimura theory for mock modular forms (2013, result): “The recent works of Zwegers [29] [30] on Ramanujan's mock theta functions, combined with the important seminal paper of Bruinier and Funke [5], have catalyzed considerable research on harmonic Maass forms (see § 2 for the definition and basic facts).”
Heegner divisors, $L$-functions and harmonic weak Maass forms (2007, background): “Such forms have been a source of recent interest due to their connection to Ramanujan’s mock theta functions (see [BO1, BO2, BOR, O2, Za3, Zw1, Zw2 ]).”
RANK DIFFERENCES FOR OVERPARTITIONS (2007, background): “In general, the generalized Lambert series which arise in the study of rank differences are known to be building blocks of mock theta functions [ 26 ].”
Asymptotics for rank and crank moments (2009, background): “Ramanujan’s famous mock theta functions are prime examples of this phenomenon [27, 29].”
Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols (2008, background): “In light of recent work of the rst author and Ono [13, 14] combined with work of Zwegers [32], [ 33 ] we now know that what Ramanujan called mock theta functions in his last letter to Hardy [30] are actually \holomorphic parts" of weak Maass forms.”
Notes on the K3 Surface and the Mathieu Group M 24 (2010, methodology): “In such a situation we can use the method recently developed by mathematicians [9, 10, 11] and construct the PoincaréMaas series to derive the above asymptotic formula.”
Eulerian series as modular forms (2007, methodology): “where the equality follows from the integral identity (see [ 22 ])”
Dyson’s ranks and Maass forms (2010, methodology): “we employ are analogous to those employed by Zwegers in his work on Ramanujan’s mock theta functions (for example, see Section 3 of [35], or [ 36 ]).”
Cycle integrals of the j-function and mock modular forms (2011, background): “This is mostly due to the discovery of Zwegers [47], [46] that Ramanujan’s mock theta functions can be completed to become modular by the addition of a certain nonholomorphic function on the upper half plane H.”