Higher level $q$-multiple zeta values with applications to quasimodular forms and partitions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916632416223232 |
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| author | Craig, William |
| author_facet | Craig, William |
| contents | In recent years, the generalized sum-of-divisor functions of MacMahon have been unified into the algebraic framework of $q$-multiple zeta values. In particular, these results link partition theory, quasimodular forms, $q$-multiple zeta values, and quasi-shuffle algebras. In this paper, we complete this idea of unification for higher levels, demonstrating that any quasimodular form of weight $k \geq 2$ and level $N$ may be expressed in terms of the $q$-multiple zeta values of level $N$ studied algebraically by Yuan and Zhao. We also give results restricted to $q$-multiple zeta values with integer coefficients, and we construct completely additive generating sets for spaces of quasimodular forms and for quasimodular forms with integer coefficients. We also provide a variety of computational examples from number-theoretic perspectives that suggest many new applications of the algebraic structure of $q$-multiple zeta values to quasimodular forms and partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13874 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher level $q$-multiple zeta values with applications to quasimodular forms and partitions Craig, William Number Theory Combinatorics In recent years, the generalized sum-of-divisor functions of MacMahon have been unified into the algebraic framework of $q$-multiple zeta values. In particular, these results link partition theory, quasimodular forms, $q$-multiple zeta values, and quasi-shuffle algebras. In this paper, we complete this idea of unification for higher levels, demonstrating that any quasimodular form of weight $k \geq 2$ and level $N$ may be expressed in terms of the $q$-multiple zeta values of level $N$ studied algebraically by Yuan and Zhao. We also give results restricted to $q$-multiple zeta values with integer coefficients, and we construct completely additive generating sets for spaces of quasimodular forms and for quasimodular forms with integer coefficients. We also provide a variety of computational examples from number-theoretic perspectives that suggest many new applications of the algebraic structure of $q$-multiple zeta values to quasimodular forms and partitions. |
| title | Higher level $q$-multiple zeta values with applications to quasimodular forms and partitions |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2409.13874 |