Mock modular Eisenstein series with Nebentypus

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions $\sigma_{\psi}^{\mathrm{sm}}(n)$. More precisely, in terms of the weight 2 quasimodular Eisenstein series $E_2(\tau)$ and a generic Shimura theta function $\theta_{\psi}(\tau)$, we show that there is a constant $\alpha_{\psi}$ for which $$\mathcal{E}^{+}_{\psi}(\tau):= \alpha_{\psi}\cdot\frac{E_2(\tau)}{\theta_{\psi}(\tau)}+ \frac{1}{\theta_{\psi}(\tau)} \sum_{n=1}^\infty \sigma^{\mathrm{sm}}_\psi(n)q^n$$ is a half integral weight (polar) mock modular form. These include generating functions for combinatorial objects such as the Andrews $spt$-function and the "consecutive parts" partition function. Finally, in analogy with Serre's result that the weight $2$ Eisenstein series is a $p$-adic modular form, we show that these forms possess canonical congruences with modular forms.