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The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

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Abstract

We study the algebra \({{\mathrm{{\mathcal {MD}}}}}\) of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in \({\mathbb {Q}}\) arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra \({{\mathrm{{\mathcal {MD}}}}}\) is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in \({{\mathrm{{\mathcal {MD}}}}}\). The (quasi-)modular forms for the full modular group \({{\mathrm{SL}}}_2({\mathbb {Z}})\) constitute a subalgebra of \({{\mathrm{{\mathcal {MD}}}}}\), and this also yields linear relations in \({{\mathrm{{\mathcal {MD}}}}}\). Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those of length 2, coming from modular forms.

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Notes

  1. In [10] certain linear combinations of these functions were called combinatorial Eisenstein series.

  2. The brackets \([2,\ldots ,2]\) were in the context of partitions already studied by P.A. MacMahon (see [16]) and named generalized divisor sums. It was shown in [1] that these are quasi-modular forms, see also Remark 2.1.

  3. The total running time on a standard PC for each entry was less than 24 h. We point to the fact that refinements of our code may give some more entries in the table.

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Acknowledgments

We thank O. Bouillot, F. Brown, J. Burgos, H. Gangl, O. Schnetz, D. Zagier, J. Zhao and W. Zudilin for their interest in our work and for helpful remarks.

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  1. Fachbereich Mathematik (AZ), Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Henrik Bachmann & Ulf Kühn

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  1. Henrik Bachmann
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Correspondence to Ulf Kühn.

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Bachmann, H., Kühn, U. The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. Ramanujan J 40, 605–648 (2016). https://doi.org/10.1007/s11139-015-9707-7

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