Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

William Craig

TL;DR

The paper extends the unification of MacMahon-type divisor sums with $q$-multiple zeta values to higher level $N$, showing that every quasimodular form of weight $k\ge2$ and level $N$ can be expressed as twists by $\zeta_N$ of level-$N$ $q$-multiple zeta values with highest weight and depth $k$. It develops integral-coefficient variants, establishes additive generating sets, and leverages the quasi-shuffle algebra of level-$N$ $q$-MZVs to derive explicit product and trace formulas. Applications include linearization of Atkin–Garvan crank moments, prime-detecting partition functions in arithmetic progressions, and representations of Ramanujan’s $\tau$-function and quadratic forms within the $q$-MZV framework. The results illuminate a rich algebraic structure connecting quasimodular forms, partitions, and $q$-series at higher level, with numerous computational examples and open questions guiding future work.

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 $k \geq 2$ 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.

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

TL;DR

The paper extends the unification of MacMahon-type divisor sums with -multiple zeta values to higher level , showing that every quasimodular form of weight and level can be expressed as twists by of level- -multiple zeta values with highest weight and depth . It develops integral-coefficient variants, establishes additive generating sets, and leverages the quasi-shuffle algebra of level- -MZVs to derive explicit product and trace formulas. Applications include linearization of Atkin–Garvan crank moments, prime-detecting partition functions in arithmetic progressions, and representations of Ramanujan’s -function and quadratic forms within the -MZV framework. The results illuminate a rich algebraic structure connecting quasimodular forms, partitions, and -series at higher level, with numerous computational examples and open questions guiding future work.

Abstract

In recent years, the generalized sum-of-divisor functions of MacMahon have been unified into the algebraic framework of -multiple zeta values. In particular, these results link partition theory, quasimodular forms, -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 and level may be expressed in terms of the -multiple zeta values of level studied algebraically by Yuan and Zhao. We also give results restricted to -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 -multiple zeta values to quasimodular forms and partitions.
Paper Structure (26 sections, 27 theorems, 121 equations)

This paper contains 26 sections, 27 theorems, 121 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Nuts and Bolts
  3. Quasimodular forms for congruence subgroups
  4. Quasimodular forms in level 1
  5. Quasimodular forms in level $N$
  6. Basics of Eisenstein series of level $N$
  7. Generation by Eisenstein series of level $N$
  8. Quasi-shuffle algebras
  9. $q$-multiple zeta values
  10. Level one $q$-multiple zeta values
  11. Level $N$ $q$-multiple zeta values
  12. $q$-multiple zeta values with integral coefficients
  13. Quasi-shuffle algebras and $q$-multiple divisor sums
  14. Proof of Main Results
  15. Proof of Theorem \ref{['T: Main Theorem, Cyclotomic']}
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $f$ be a (mixed weight) quasimodular form of (maximal) weight $k \geq 1$ for the congruence subgroup $\Gamma(N)$, $N \geq 3$. Then if $k \geq 2$ or if $k=1$ and $f$ belongs to the Eisenstein space $\mathcal{E}_1\left( \Gamma(N) \right)$, $f$ can be represented as a linear combination of twists b

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Theorem 1.3
  • Remark
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3: 123
  • Lemma 2.4
  • ...and 27 more