Abstract
In 1919, P. A. MacMahon studied generating functions for generalized divisor sums. In this paper, we provide a framework in which to view these generating functions in terms of Jacobi forms, and prove that they are quasi-modular forms.
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References
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MacMahon, PA: Divisors of numbers and their continuations in the theory of partitions. In: Andrews, G (ed.)Reprinted: Percy A. MacMahon Collected Papers, pp. 305–341. MIT Press, Cambridge (1986).
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Acknowledgements
The author would like to thank the Max-Planck-Institut für Mathematik for hosting him while this research was being prepared. Furthermore, he would also like to thank Kathrin Bringmann, Robert Osburn, Noriko Yui, and Don Zagier for fruitful discussions about this topic, as well as the referees for their helpful comments.
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Rose, S.C. Quasi-modularity of generalized sum-of-divisors functions. Res. number theory 1, 18 (2015). https://doi.org/10.1007/s40993-015-0019-1
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DOI: https://doi.org/10.1007/s40993-015-0019-1