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We study the algebra MD of generating function for multiple divisor sums andits connections to multiple zeta values. The generating functions for multipledivisor sums are formal power series in q with coefficients in Q arising fromthe calculation of the Fourier expansion of multiple Eisenstein series. We showthat the algebra MD is a filtered algebra equipped with a derivation and usethis derivation to prove linear relations in MD. The (quasi-)modular forms forthe full modular group Sl_2(Z) constitute a sub-algebra of MD this also yieldslinear relations in MD. Generating functions of multiple divisor sums can beseen as a q-analogue of multiple zeta values. Studying a certain map from thisalgebra into the real numbers we will derive a new explanation for relationsbetween multiple zeta values, including those in length 2, coming from modularforms.

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