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Related Experiment Video

Updated: Jun 24, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
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

Proceedings of the National Academy of Sciences of the United States of America
|April 2, 2009
PubMed
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.

Related Experiment Videos

Last Updated: Jun 24, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

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.