  Given a unimodular lattice $\Lambda\subseteq \mathbb{R}^2$ consider the
counting function $\mathcal{N}_\Lambda(T)$ counting the number of lattice
points of norm less than $T$, and the remainder
$\mathcal{R}_\Lambda(T)=\mathcal{N}(T)-\pi T^2$. We give an elementary proof
that the mean square of the remainder over the set of all shears of a
unimodular lattice is bounded by $O(T\log^2(T))$.
