Mertens, Michael H, Ono, Ken and Rolen, Larry
(2021)
Mock modular Eisenstein series with Nebentypus.
International Journal of Number Theory, 17 (03).
pp. 683-697.
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Abstract
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.
| Item Type: | Article |
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| Additional Information: | 11 pages, v2: corrected small error; v3: minor edits in response to referee's report; v4: accepted version |
| Uncontrolled Keywords: | math.NT, math.NT |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 08 Feb 2021 08:53 |
| Last Modified: | 18 Jan 2023 23:01 |
| DOI: | 10.1142/S179304212040028X |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3115275 |
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