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.