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

Abstract

We consider the t-hook functions on partitions $f_{a,t}: \mathcal {P}\rightarrow \mathbb {C}$ defined by

$$\begin{aligned} f_{a,t}(\lambda ):=t^{a-1} \sum _{h\in \mathcal {H}_t(\lambda )}\frac{1}{h^a}, \end{aligned}$$

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 $a\ge 2$, we show that these q-brackets are natural pieces of weight $2-a$ sesquiharmonic and harmonic Maass forms, while for odd $a\le -1,$ 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.