Eichler integrals of Eisenstein series as $q$-brackets of weighted $t$-hook functions on partitions

TL;DR

This paper explores the connection between $q$-brackets of $t$-hook functions on partitions and Eichler integrals of Eisenstein series, revealing their roles as Maass forms or quantum modular forms depending on parameters.

Contribution

It establishes that $q$-brackets of $t$-hook functions are related to Maass forms or quantum modular forms, providing new formulas and asymptotic expansions involving special number-theoretic functions.

Findings

For even $a\, extgreater= 2$, $q$-brackets are parts of Maass forms.

For odd $a\, extless= -1$, $q$-brackets are holomorphic quantum modular forms.

Derived new formulas of Chowla-Selberg type and asymptotics involving zeta and Bernoulli numbers.

Abstract

We consider the $t$-hook functions on partitions $f_{a,t}: \mathcal{P}\rightarrow \mathbb{C}$ defined by $$ f_{a,t}(\lambda):=t^{a-1} \sum_{h\in \mathcal{H}_t(\lambda)}\frac{1}{h^a}, $$ where $\mathcal{H}_t(\lambda)$ is the multiset of partition hook numbers that are multiples of $t$. The Bloch-Okounkov $q$-brackets $\langle f_{a,t}\rangle_q$ include Eichler integrals of the classical Eisenstein series. For even $a\geq 2$, we show that these $q$-brackets are natural pieces of weight $2-a$ sesquiharmonic and harmonic Maass forms, while for odd $a\leq -1,$ we show that they are holomorphic quantum modular forms. We use these results to obtain new formulas of Chowla-Selberg type, and asymptotic expansions involving values of the Riemann zeta-function and Bernoulli numbers. We make use of work of Berndt, Han and Ji, and Zagier.