Overpartition $M2$-rank differences, class number relations, and vector-valued mock Eisenstein series
Volume 189 / 2019
Acta Arithmetica 189 (2019), 347-365
MSC: Primary 11F37; Secondary 11E41, 11F27.
DOI: 10.4064/aa170810-21-10
Published online: 7 June 2019
Abstract
We prove that the generating function of overpartition $M2$-rank differences is, up to coefficient signs, a component of the vector-valued mock Eisenstein series attached to a certain quadratic form. We use this to compute analogs of the class number relations for $M2$-rank differences. As applications we split the Kronecker–Hurwitz relation into its “even” and “odd” parts and calculate sums over Hurwitz class numbers of the form $\sum_{r \in \mathbb{Z}} H(n - 2r^2)$.
