sum of square of cube function $r_3(n)$

Question

I want to know that from MSE on sum of squares of $r_2(n)$, is it possible to have something like, ,where $r_3(n)$ denotes the number of ways $n=a^2+b^2+c^2$ for integers $a,b,c$ $$\sum_{n\le\mathcal N}r_3^2(n)\sim C\mathcal N^{3/2}\log^c\mathcal N$$ for some constant $c$? I think with the same argument with $3$ dimensional sphere (containing lattice points) we have something like the following? But I wonder about the case above $$\sum_{n\le \mathcal N}r_3(n)\sim \pi\mathcal N^{3/2}$$