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Nonholomorphic Ramanujan-type congruences for Hurwitz class numbers
Olivia Beckwith1, Martin Raum2, Olav K Richter3
1Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801.
Summary
Ramanujan-type congruences for Hurwitz class numbers are shown to work with nonholomorphic generating series, a novel finding. This study establishes a divisibility result for these unique nonholomorphic congruences.
Area of Science:
- Number Theory
- Modular Forms
- Algebraic Geometry
Background:
- Ramanujan-type congruences are typically associated with holomorphic generating series.
- Hurwitz class numbers are significant arithmetic objects with connections to modular forms.
Purpose of the Study:
- To investigate the existence and properties of Ramanujan-type congruences for Hurwitz class numbers using nonholomorphic generating series.
- To establish a divisibility result for these newly discovered nonholomorphic congruences.
Main Methods:
- Utilizing the holomorphic projection of products of theta series with a Hurwitz class number generating series.
- Applying a theorem by Serre to exclude certain congruence possibilities.
Main Results:
- Demonstrated that Ramanujan-type congruences for Hurwitz class numbers can be supported by nonholomorphic generating series.
- Established a specific divisibility result for these nonholomorphic congruences.
Conclusions:
- This research expands the understanding of Ramanujan-type congruences beyond holomorphic settings.
- The findings introduce a new class of congruences for Hurwitz class numbers with potential implications in number theory.
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