  We study the algebra MD of generating function 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 sub-algebra of MD 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 in length 2, coming from modular
forms.
