Holomorphic projections and Ramanujan's mock theta functions

Area of Science:

• Number Theory
• Harmonic Analysis
• Automorphic Forms

Background:

• Vector-valued harmonic weak Maass forms and vector-valued modular forms are complex mathematical objects with deep connections to number theory.
• Ramanujan's mock theta functions are enigmatic series with significant implications in various mathematical fields.

Purpose of the Study:

• To establish a novel holomorphic projection operator for tensor products of specific types of automorphic forms.
• To leverage this operator to uncover new, simple recurrence relations for the Fourier coefficients of Ramanujan's mock theta functions.

Main Methods:

• Utilizing spectral methods within the theory of automorphic forms.
• Constructing a holomorphic projection operator acting on tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms.

Main Results:

• Successfully established a holomorphic projection operator.
• Discovered simple recursions for the Fourier series coefficients of Ramanujan's mock theta functions.

Conclusions:

• The developed holomorphic projection operator provides a powerful tool for studying automorphic forms.
• The discovered recursions offer new insights into the structure and properties of Ramanujan's mock theta functions.