Lattice points in many-dimensional balls

Let $P_k(n)$ be the difference of the number of points of the integer lattice contained in the ball $y_1^2+\dots+y_k^2\leq n$ and the volume of this ball. We investigate the asymptotic behavior of the sums $\sum_{n\leq x}P_k(n)$, $(k\geq4)$, $\sum_{n\leq x}P_3^2(n)$, and $\sum_{n\leq x}P_4^2(n)$ as $x\to+\infty$.