A modular framework for generalized Hurwitz class numbers I

Abstract

We discover a non-trivial relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers. This relation yields a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$, where $N > 1$ is stipulated to be odd and square-free. We extend this observation to a non-holomorphic framework and obtain a higher level non-holomorphic Zagier Eisenstein series as well as a preimage $\mathcal{G}$ of it under the differential operator $\xi_{\frac{1}{2}}$. All of these observations are deduced from a more general inspection of a certain weight $\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point $s=\frac{3}{4}$. This idea goes back to Duke, Imamo\={g}lu and T\'{o}th in level $4$ and relies on the theory of so-called sesquiharmonic Maass forms. We calculate the Fourier expansion of $\mathcal{G}$ and $\xi_{\frac{1}{2}}\mathcal{G}$. We conclude by offering examples if $N=5$ or $N=7$ as well as some questions for future work.