Mock Eisenstein series associated to partition ranks

TL;DR

The paper addresses the rank statistics of partitions by introducing a new family of mock Eisenstein series $f_k$ whose partition traces encode the rank moments $R_k(q)$. It develops quasi-completions $f_k^*$ with modular-type transformation laws and proves integrality of the Fourier coefficients after a constant shift, together with a holomorphic anomaly equation for the completions. A recursive framework using auxiliary polynomials $g_\ell$ expresses $f_k$ in terms of traces and shows the algebra generated by $\{f_k\}$ and classical Eisenstein series is closed under the differential operator $D$ and compatible with raising/lowering operators. The results provide a structured, modular-analytic viewpoint on partition ranks, give explicit recursion and integrality statements, and open avenues toward a broader theory of higher-level mock Eisenstein series with potential arithmetic and combinatorial applications.

Abstract

In this paper, we introduce a new class of mock Eisenstein series, describe their modular properties, and write the partition rank generating function in terms of so-called partition traces of these. Moreover, we show the Fourier coefficients of the mock Eisenstein series are integral and we obtain a holomorphic anomaly equation for their completions.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3