Traces of partition Eisenstein series and almost holomorphic modular forms
Kathrin Bringmann1, Badri Vishal Pandey1
1Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany.
Research in Number Theory
|April 28, 2025
Summary
Researchers explored traces of partition Eisenstein series, deriving explicit formulas for functions. They identified the precise mathematical spaces for these functions and their modular completions, revealing relationships via operators.
Area of Science:
- Number Theory
- Representation Theory
- Harmonic Analysis
Background:
- Recent work by Amdeberhan, Griffin, Ono, and Singh introduced the study of traces of partition Eisenstein series.
- These traces have been utilized to derive explicit formulas for various mathematical functions.
- Understanding the properties and relationships of these functions is crucial in number theory and related fields.
Purpose of the Study:
- To determine the precise function spaces inhabited by the traces of partition Eisenstein series.
- To find modular completions for these series and associated functions.
- To elucidate the relationships between these objects through the application of specific operators.
Main Methods:
- Analysis of the spectral properties of Eisenstein series.
- Application of modular forms theory and related transformations.
- Investigation of linear and non-linear operators acting on these function spaces.
Main Results:
- Precise characterization of the function spaces containing the traces of partition Eisenstein series.
- Construction of modular completions, extending the domain of these functions.
- Demonstration of interrelations between these functions and their completions via identified operators.
Conclusions:
- The study provides a rigorous framework for understanding traces of partition Eisenstein series.
- The identified relationships offer new insights into the structure of automorphic forms and related L-functions.
- This work bridges concepts from number theory, representation theory, and harmonic analysis.
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