  The Gauss Circle Problem concerns finding asymptotics for the number of
lattice point lying inside a circle in terms of the radius of the circle. The
heuristic that the number of points is very nearly the area of the circle is
surprisingly accurate. In this work, we describe two variants of the Gauss
Circle problem that exhibit similar characteristics.
  The first variant concerns sums of Fourier coefficients of $\text{GL}(2)$
cusp forms. These sums behave very similarly to the error term in the Gauss
Circle problem. We introduce new Dirichlet series with coefficients that are
squares of partial sums of Fourier coefficients of cusp forms. We study the
meromorphic properties of these Dirichlet series and use these series to give
new perspectives on the mean square of the size of sums of these Fourier
coefficients.
  The second variant concerns the number of lattice points of bounded size on
one-sheeted hyperboloids. This problem is very similar to counting the number
of lattice points within a spheres of various dimensions, except with the
additional constraint of lying on a hyperboloid. It turns out that this problem
is equivalent to estimating sums of the shape $\sum r_d(n^2 + h)$, where
$r_d(m)$ is the number of representations of $m$ as a sum of $d$ squares. We
prove improved smoothed and sharp estimates of the second moment of these sums,
yielding improved estimates of the number of lattice points.
