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Macmahon’s sums-of-divisors and their connection to multiple Eisenstein series

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Abstract

We give explicit expressions for MacMahon’s generalized sums-of-divisors q-series \(A_r\) and \(C_r\) by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular forms, vertex algebras, \(N=4\) SU(N) Super–Yang–Mills theory, and the study of congruences of partitions. We relate them to a broader mathematical framework and give explicit expressions for both q-series in terms of Eisenstein series and their odd variants.

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Notes

  1. In the case \(k_1=2\) one needs to use Eisenstein summation.

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Acknowledgements

This project was partially supported by JSPS KAKENHI Grant 23K03030. The author would like to thank Sven Möller for making him aware of the appearances of q-analogues of multiple zeta values in the context of vertex algebras and the work [4]. In addition he would like to thank Jan-Willem van Ittersum for comments on the first version of this paper and the anonymous referee for corrections.

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  1. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Henrik Bachmann

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Bachmann, H. Macmahon’s sums-of-divisors and their connection to multiple Eisenstein series. Res. number theory 10, 50 (2024). https://doi.org/10.1007/s40993-024-00537-2

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