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The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$.

Is there some generalization of a modular form (quasimodular, mock modular, etc.) where you can find some similar Fourier expansion of the form $$f(z) = 1 + C \sum_{n=0}^{\infty} \sigma_0(n) q^n, \; q = e^{2\pi i z}$$ involving just the number-of-divisors function $\sigma_0(n)$? I haven't been able to find this.

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  • $\begingroup$ Do you know of examples of generalized modular forms involving any $\sigma_{k-1}$ where $k - 1$ is even? $\endgroup$
    – Vincent
    Jul 23, 2014 at 8:19
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    $\begingroup$ This question was asked before, and I answered it here: mathoverflow.net/questions/128533/… $\endgroup$
    – GH from MO
    Jul 23, 2014 at 8:43
  • $\begingroup$ @GHfromMO: Why don't you want the question to be closed as duplicate then(?) -- At least so far you didn't cast a close vote. $\endgroup$
    – Stefan Kohl ♦
    Jul 23, 2014 at 15:20
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    $\begingroup$ @StefanKohl: Done. $\endgroup$
    – GH from MO
    Jul 23, 2014 at 15:29

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