Overpartition $M2$-rank differences, class number relations, and vector-valued mock Eisenstein series

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)$.