Speaker

Ken Ono (University of Virginia)

Date

October 8 8:30-9:30, Japan (October 7, 19:30-20:30 Virginia, USA)

Title

Eichler integrals of Eisenstein series as \(q\)-brackets and various types of modular forms

Abstract

We consider the \(t\)-hook functions on partitions \(f_{a,t}: \mathcal{P}\rightarrow \mathbb{C}\) defined by \begin{align} f_{a,t}(\lambda):=t^{a-1} \sum_{h\in \mathcal{H}_t(\lambda)}\frac{1}{h^a}, \end{align} 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 positive even integers \(a\), we complete these \(q\)-brackets to obtain weight \(2-a\) sesquiharmonic and harmonic Maass forms. For odd \(a\leq -1,\) we show that these \(q\)-brackets are holomorphic quantum modular forms. Using these results, we obtain partition theoretic formulas of Chowla-Selberg type, and asymptotic expansions involving values of the Riemann zeta-function and Bernoulli numbers. These results are obtained using work of Berndt, Han and Ji, and Zagier. This is joint work with Kathrin Bringmann and Ian Wagner.