2024/09/20 by Craig, William · 3 citations
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.13874
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 ≥ 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.