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The Ramanujan Journal

, Volume 40, Issue 3, pp 605–648 | Cite as

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

  • Henrik Bachmann
  • Ulf Kühn
Article

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.

Keywords

Multiple zeta values q-Analogues of multiple zeta values (quasi-)modular forms Multiple divisor sums Quasi-shuffle algebras  Multiple Eisenstein series 

Mathematics Subject Classification

11M32 11F11 13J05 33E20 05A30 

Notes

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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Fachbereich Mathematik (AZ)Universität HamburgHamburgGermany

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