  Generalised Eisenstein series are non-holomorphic modular invariant functions
of a complex variable, $\tau$, subject to a particular inhomogeneous Laplace
eigenvalue equation on the hyperbolic upper-half $\tau$-plane. Two infinite
classes of such functions arise quite naturally within different string theory
contexts. A first class can be found by studying the coefficients of the
effective action for the low-energy expansion of type IIB superstring theory,
and relatedly in the analysis of certain integrated four-point functions of
stress tensor multiplet operators in $\mathcal{N} = 4$ supersymmetric
Yang-Mills theory. A second class of such objects is known to contain all
two-loop modular graph functions, which are fundamental building blocks in the
low-energy expansion of closed-string scattering amplitudes at genus one. In
this work, we present a Poincar\'e series approach that unifies both classes of
generalised Eisenstein series and manifests certain algebraic and differential
relations amongst them. We then combine this technique with spectral methods
for automorphic forms to find general and non-perturbative expansions at the
cusp $\tau \to i \infty$. Finally, we find intriguing connections between the
asymptotic expansion of these modular functions as $\tau \to 0$ and the
non-trivial zeros of the Riemann zeta function.
