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

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.