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

We study the algebra 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 Q arising from the calculation of the Fourier expansion of multiple Eisenstein s...

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Bibliographic Details
Published in:The Ramanujan journal 2016-08, Vol.40 (3), p.605-648
Main Authors: Bachmann, Henrik, Kühn, Ulf
Format: Article
Language:English
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
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Summary:We study the algebra 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 Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD . The (quasi-)modular forms for the full modular group SL 2 ( Z ) constitute a subalgebra of MD , and this also yields linear relations in 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.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-015-9707-7