On the distribution of integral points on cones

Let $r_k(n)$ denote the number of representations of a positive integer $n$ as the sum of $k$ squares. We prove that
$$ \sum_{n\le x}r^2_3(n)=Cx^2+O\Big(x^\frac32\big(\log x\big)^\frac72\Big), $$
where $C>0$ is a certain constant, and that
$$ \sum_{n\le x}r^2_4(n)=32\zeta(3)x^3+O\Big(x^2\big(\log x\big)^\frac53\Big). $$
Bibl. 14 titles.