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<p>In recent years, the generalized sum-of-divisor functions of MacMahon have been unified into the algebraic framework of <i>q</i>-multiple zeta values. In particular, these results link partition theory, quasimodular forms, <i>q</i>-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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and level <i>N</i> may be expressed in terms of the <i>q</i>-multiple zeta values of level <i>N</i> studied algebraically by Yuan and Zhao. We also give results restricted to <i>q</i>-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 <i>q</i>-multiple zeta values to quasimodular forms and partitions.</p>

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Higher level q-multiple zeta values with applications to quasimodular forms and partitions

  • 期刊论文

  • 发表日期：

    2026年7月17日

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  • William Craig

摘要

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 \(k \ge 2\) k ≥ 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.