  The coefficients of the higher-derivative terms in the low energy expansion
of genus-one graviton Type II superstring scattering amplitudes are determined
by integrating sums of non-holomorphic modular functions over the complex
structure modulus of a torus. In the case of the four-graviton amplitude, each
of these modular functions is a multiple sum associated with a Feynman diagram
for a free massless scalar field on the torus. The lines in each diagram join
pairs of vertex insertion points and the number of lines defines its weight
$w$, which corresponds to its order in the low energy expansion. Previous
results concerning the low energy expansion of the genus-one four-graviton
amplitude led to a number of conjectured relations between modular functions of
a given $w$, but different numbers of loops $\le w-1$. In this paper we shall
prove the simplest of these conjectured relations, namely the one that arises
at weight $w=4$ and expresses the three-loop modular function $D_4$ in terms of
modular functions with one and two loops. As a byproduct, we prove three
intriguing new holomorphic modular identities.
