Gauss Sphere Problem with Polynomials

This paper provides an estimate of the sum of a homogeneous polynomial $P$ of degree $\nu$ and mean zero over the lattice points inside a sphere of radius $R$. It is proved that $$\sum _{\genfrac{}{}{0px}{}{\mathbf{x}\in {\mathbb{Z}}^3}{\mathrm{\mid }\mathbf{x}\mathrm{\mid }\le R}}P(\mathbf{x})=O_{ϵ,P}(R^{\nu +83\mathrm{/}64+ϵ})$$