Eisenstein-type series associated to partition ranks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915798927278080 |
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| author | Bringmann, Kathrin Pandey, Badri Vishal van Ittersum, Jan-Willem |
| author_facet | Bringmann, Kathrin Pandey, Badri Vishal van Ittersum, Jan-Willem |
| contents | In this paper, we introduce a class of functions that behave like classical Eisenstein series in many ways, but with a key distinction: only their non-holomorphic completions transform like (quasi)modular forms. We show how the partition rank generating function can be expressed in terms of partition traces of these functions. A key feature of our construction is that the completions satisfy a holomorphic anomaly equation - a phenomenon typically seen in the context of quantum field theory and string theory. We also show that the Fourier coefficients of these Eisenstein-type series are integral. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eisenstein-type series associated to partition ranks Bringmann, Kathrin Pandey, Badri Vishal van Ittersum, Jan-Willem Number Theory Mathematical Physics Combinatorics 11F03, 11F11, 11F37, 11F50, 11P82 In this paper, we introduce a class of functions that behave like classical Eisenstein series in many ways, but with a key distinction: only their non-holomorphic completions transform like (quasi)modular forms. We show how the partition rank generating function can be expressed in terms of partition traces of these functions. A key feature of our construction is that the completions satisfy a holomorphic anomaly equation - a phenomenon typically seen in the context of quantum field theory and string theory. We also show that the Fourier coefficients of these Eisenstein-type series are integral. |
| title | Eisenstein-type series associated to partition ranks |
| topic | Number Theory Mathematical Physics Combinatorics 11F03, 11F11, 11F37, 11F50, 11P82 |
| url | https://arxiv.org/abs/2504.07713 |