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Updated: Jun 24, 2026
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
Overpartitions and class numbers of binary quadratic forms
Kathrin Bringmann1, Jeremy Lovejoy
1Mathematisches Institut, Universität Köln, Weyertal 86-90, D-50931 Cologne, Germany. kbringma@math.uni-koeln.de
Summary
The Zagier-Eisenstein series
Area of Science:
- Number Theory
- Combinatorics
Background:
- Zagier-Eisenstein series are important objects in number theory.
- Overpartitions and their ranks are key combinatorial objects.
- Weak Maass forms connect number theory and analysis.
Purpose of the Study:
- To explore the relationship between Zagier-Eisenstein series and weak Maass forms.
- To investigate the properties of overpartition rank differences.
- To derive new identities and congruences for these mathematical objects.
Main Methods:
- Analyzing the nonholomorphic and holomorphic parts of series.
- Utilizing generating functions for combinatorial quantities.
- Applying techniques from the theory of modular forms.
Main Results:
- The nonholomorphic part of the Zagier-Eisenstein series matches that of specific weak Maass forms.
- The holomorphic parts of these weak Maass forms generate functions for overpartition rank differences.
- New exact formulas, asymptotic formulas, and congruences for overpartition rank differences were obtained.
Conclusions:
- This study reveals a deep connection between Zagier-Eisenstein series and weak Maass forms.
- The findings provide significant insights into the structure and properties of overpartition rank differences.
- The results lead to novel q-series identities of the mock theta type.
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