Explicit bounds for counting integer points inside $2$-norm balls

Question

It is well known that a ball of $2$ norm radius $R$ in $d$ dimensional space constains \begin{align} N(R) = V_d R^d + \mathcal{O}(R^{d-2}) \end{align} points with integer coordinates (i.e. elements of $\mathbb{Z}^d$ embedded in $R^d$ in the obvious way), as long as $d\geq 4$. Here $V_d$ is the volume of the $d$ dimensional unit ball. I believe this result follows from the $4$ square theorem. My question is can we obtain bounds of the form \begin{align} \left\lvert{N(R) - V_d R^d}\right\rvert \leq a_d R^{d-2}. \end{align} or, if this is too difficult, something like \begin{align} b_d R^d \leq N(R) \leq c_d R^d, \end{align} for some reasonable $a_d$ and $b_d$ and $c_d$?