Quasi-modularity in MacMahon partition variants and prime detection

TL;DR

The paper develops a unified quasi-modular framework for MacMahon partition variants by introducing generalized MacMahon functions $A_{S,N,\epsilon,k}(q)$ and showing they are expressible as Lehmer polynomials in Eisenstein-type series $G_{S,N,\epsilon,2j}(q)$. When the index set $S$ is symmetric, these functions are quasi-modular on appropriate congruence subgroups, recovering and extending Rose–Larson-type results. The authors extract explicit level-2 prime-detecting expressions for $M_k^{(2)}(n)$ via Lelièvre's criteria and provide a lattice-theoretic analogue using $r_L(n)$ for $L\in\{L_1,L_2,E_8\}$ to detect primes in arithmetic progressions modulo $4$. They also discuss obstructions from cusp forms that limit further prime-detecting expressions and outline how these methods unify various MacMahon variants. The work offers concrete tools for prime-detection grounded in modular and lattice-theoretic structures with potential extensions to higher levels and related partition functions.

Abstract

Building on the results of Craig, van Ittersum, and Ono, we provide a refined understanding of MacMahon's partition functions and their variants, including their quasi-modular properties and new prime-detecting expressions.

Figures (1)

• Figure 1: Two examples of partitions of 17 squares into 3 rectangular blocks: $1 \times 1$, $2 \times 3$, $5 \times 2$ and $2 \times 1$, $3 \times 1$, $4 \times 3$. In these cases, the products of the heights are given by $1 \cdot 3 \cdot 2$ and $1 \cdot 1 \cdot 3$, respectively.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Proposition 2.2: KanekoZagier1995 and Zagier2008
• Definition 3.1
• Definition 3.2
• Remark 1
• Definition 3.3
• Theorem 3.4: A generalized version of \ref{['thm:MacMahon-Lehmer']}