Modularity of moments of reciprocal sums for partitions into distinct parts

TL;DR

The paper investigates the modularity properties of generating functions for the moments $s_k(n)$ of reciprocal sums over partitions into distinct parts. It constructs modular completions $\widehat{g}_k(\tau)$, expressing them in terms of Maass Eisenstein series and Eichler integrals: $\widehat{g}_1$ is a weight-0 sesquiharmonic Maass form on $\Gamma_0(2)$ with shadow linked to $\widehat{E}_2$, while for $k\ge2$, $\widehat{g}_k$ is a Maass form on $\Gamma_0(2)$ of eigenvalue $k(1-k)$ and has holomorphic part $g_k(q)$. In particular, the explicit case $k=2$ yields $\widehat{g}_2(\tau)=\frac{\pi^2}{90}\bigl(E(\tau;2)-4E(2\tau;2)\bigr)$, and the holomorphic part recovers $g_2(q)$, with connections to $s_3^*(n)$. The results reveal a rich modular structure in combinatorial generating functions for partitions and offer tools, via Kronecker limit formulas and Eichler integrals, to sharpen asymptotics and explore twisted variants of reciprocal-sum moments.

Abstract

In this paper, we determine modularity properties of the generating function of $s_k(n)$ which sums $k$-th power of reciprocals of parts throughout all of the partitions of $n$ into distinct parts. In particular, we show that the generating function for $s_k (n)$ is related to Maass Eisenstein series and sesquiharmonic Maass forms.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.3: BFOR
• Lemma 2.4
• Lemma 2.5
• proof : Proof of \ref{['thm:g1']}
• proof : Proof of \ref{['thm:g_1 limit']}