  Froyshov invariants are numerical invariants of rational homology
three-spheres derived from gradings in monopole Floer homology. In the past few
years, they have been employed to solve a wide range of problems in three and
four-dimensional topology. In this paper, we look at connections with
hyperbolic geometry for the class of minimal $L$-spaces. In particular, we
study relations between Froyshov invariants and closed geodesics using ideas
from analytic number theory. We discuss two main applications of our approach.
First, we derive effective upper bounds for the Froyshov invariants of minimal
hyperbolic $L$-spaces purely in terms of volume and injectivity radius. Second,
we describe an algorithm to compute Froyshov invariants of minimal $L$-spaces
in terms of data arising from hyperbolic geometry. As a concrete example of our
method, we compute the Froyshov invariants for all spin$^c$ structures on the
Seifert-Weber dodecahedral space. Along the way, we also prove several results
about the eta invariants of the odd signature and Dirac operators on hyperbolic
three-manifolds which might be of independent interest.
