  The analytic contribution to the low energy expansion of Type II string
amplitudes at genus-one is a power series in space-time derivatives with
coefficients that are determined by integrals of modular functions over the
complex structure modulus of the world-sheet torus. These modular functions are
associated with world-sheet vacuum Feynman diagrams and given by multiple sums
over the discrete momenta on the torus. In this paper we exhibit exact
differential and algebraic relations for a certain infinite class of such
modular functions by showing that they satisfy Laplace eigenvalue equations
with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein
series. Furthermore, we argue that the set of modular functions that contribute
to the coefficients of interactions up to order D**10 R*4 are linear sums of
functions in this class and quadratic polynomials in Eisenstein series and odd
Riemann zeta values. Integration over the complex structure results in
coefficients of the low energy expansion that are rational numbers multiplying
monomials in odd Riemann zeta values.
