Mock Theta Functions

TL;DR

This thesis explores the theory of mock theta functions introduced by Ramanujan, providing new interpretations within real-analytic modular forms and explicit results for several mock theta functions.

Contribution

It offers a novel interpretation of mock theta functions as real-analytic modular forms and derives explicit correction terms for multiple orders of these functions.

Findings

Mock theta functions can be completed to real-analytic modular forms.

Explicit correction terms involve period integrals of unary theta functions.

Results include explicit formulas for 8 fifth order and all 3 seventh order 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.