Mock Theta Functions
Sander Zwegers
TL;DR
This work places Ramanujan's mock theta functions in a unified framework of real-analytic modular forms. By combining Lerch sums, Mordell-type integrals, indefinite theta-functions, and meromorphic Jacobi-form Fourier coefficients, Sander Zwegers constructs explicit real-analytic completions that yield weight $1/2$ modular objects, with corrections given by period integrals of unary theta functions of weight $3/2$. The seventh- and fifth-order mock theta functions are shown to decompose as $F=H+G$, where $H$ is a weight $1/2$ real-analytic modular form (Casimir eigenvalue $ rac{3}{16}$) and $G$ is a theta-type unary sum; this provides both a modular completion and a precise structural interpretation of their non-holomorphicity. The results establish multiple routes to modularity: Lerch-sum corrections, Fourier coefficients of meromorphic Jacobi forms, and indefinite theta-function theory, and reveal deep connections between mock theta phenomena and real-analytic modular forms with period-integral representations. Overall, the thesis offers a coherent, technically detailed bridge from Ramanujan’s enigmatic mock theta functions to a robust analytic-modular framework with potential arithmetic applications.
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.