Recently, physicists such as Green, et al., have shown that the behavior of gravitons (hypothetical particles of gravity represented by massless string states) is closely related to automorphic forms. Miller, et al., have examined coefficients of the scattering amplitude of four gravitons and shown that these involve eigenvalue problems for the invariant Laplacian in different moduli spaces.  We will use functional analysis and spectral theory of automorphic forms to examine the case on $SL_2(\mathbb{Z})\backslash\mathfrak{H}$.

We also will discuss new surprising connections between these varying types of differential equations and their ties to zeros of the $\zeta$-function.