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

Abstract

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 $k\ge 2$ 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.