Mock Theta Functions

Abstract: In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms. In Chapter 1 we give results for Lerch sums (also called Appell functions, or generalized Lambert series). In Chapter 2 we consider indefinite theta functions of type (r-1,1). Chapter 3 deals with Fourier coefficients of meromorphic Jacobi forms. In Chapter 4 we use the results from Chapter 2 to give explicit results for 8 of the 10 fifth order mock theta functions and all 3 seventh order functions, that were originally defined by Ramanujan. The result is that we can find a correction term, which is a period integral of a weight 3/2 unary theta functions, such that if we add it to the mock theta function, we get a weight 1/2 real-analytic modular form, which is annihilated by the hyperbolic Laplacian.

• The paper introduces a novel framework linking mock theta functions to real-analytic modular forms via Lerch sums and indefinite theta functions.
• It employs advanced analytic techniques to establish rigorous modular transformation properties of these functions.
• The findings provide a cohesive theoretical basis that deepens our understanding of Ramanujan’s enigmatic mock theta functions.

An Examination of Mock Theta Functions

This document is an exhaustive research study centered around mock theta functions by Sander Pieter Zwegers. This particular class of mathematical functions was introduced by the esteemed mathematician Srinivasa Ramanujan in the early 20th century and has posed considerable intrigue due to their enigmatic nature and mathematical elegance. The study aims to juxtapose Ramanujan's examples of mock theta functions within a broader theoretical framework, specifically relating them to real-analytic modular forms. This connection offers a cohesive explanation for the intriguing properties of mock theta functions as indicated by Ramanujan.

Key Focus and Methodologies

The research adopts multiple paths to weave an underlying theoretical narrative for mock theta functions. It traverses through the domains of Lerch sums, indefinite theta functions associated with indefinite quadratic forms, the modulatory behavior of Fourier coefficients of meromorphic Jacobi forms, and real-analytic modular forms to effectively elucidate the enigmatic qualities of mock theta functions.

1. Lerch Sums: The research introduces and evaluates Lerch sums, a series fundamentally influenced by Lerch, which transform almost analogously to Jacobi forms. By incorporating a correction term, these sums attain exact transformation properties akin to Jacobi forms. This chapter sets the groundwork by outlining the resemblances between Lerch-like sums and mock theta functions.
2. Indefinite Theta Functions: Ascribing to quadratic forms of type (r-1, 1), this portion of the study explores defining and elucidating the properties of indefinite theta functions. These functions maintain analogous transformation properties to those of classical theta functions but are not strictly holomorphic. By formulating a revised definition, the study identifies modular and elliptic transformation properties for these functions.
3. Fourier Coefficients of Meromorphic Jacobi Forms: Zwegers extends the investigation to the modularity of Fourier coefficients of meromorphic Jacobi forms, generalizing results beyond holomorphic forms to encapsulate meromorphic distinctions.
4. Mock Theta Functions and Real-Analytic Modular Forms: Through leveraging prior results on Lerch-like sums and indefinite theta functions, the study relates mock theta functions to real-analytic modular forms, situating them in a domain that explains their properties robustly.

Impact and Theoretical Contributions

The research is vested in its novel approach to demystify several conjectures about mock theta functions and provide a theoretical structure that illuminates their properties comprehensively. It enriches the understanding of mock theta functions by aligning their characteristics with modern mathematical objects and paradigms, such as real-analytic modular forms and indefinite theta functions, fortifying the bridge between classical analysis and number theory. Moreover, the study contributes to theoretical numerics by elucidating modular transformations, thus enhancing the computational and theoretical toolkit for analyzing complex mathematical functions.

Prospects for Further Research

While the document breaks substantial ground in correlating Ramanujan's enigmatic functions with contemporary mathematical theories, certain boundaries remain for further investigation. For instance, two mock theta functions of "order 5" were not fully encapsulated within the study's framework, prompting expansions into distinct classes of indefinite quadratic forms. Additionally, future work might seek to construct an accessible spectral theory around the analytically derived modular forms.

In sum, Sander Pieter Zwegers' research significantly advances the field's comprehension of mock theta functions, providing both a cohesive narrative and groundwork for future investigative and theoretical pursuits.