Lattice points in the circle and the sphere

Let $P(x)$ and $P_3(x)$ be the error terms in the Gaussian circle problem and the sphere problem, respectively.
We investigate the asymptotic behavior of the sums
$$ \sum_{\substack{k\le x\\k\equiv0\!\!\!\pmod p}}P(k),\quad\sum_{\substack{k\le x\\k\equiv0\!\!\!\pmod p}}P_3(k). $$
Here $p\ge2$ is a prime number.