Gauss Sphere Problem with Polynomials

Abstract

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 $$\sum _{\genfrac{}{}{0ex}{}{\mathbf{x}\in {\mathbb{Z}}^3}{|\mathbf{x}|\le R}}P(\mathbf{x})=O_{\epsilon ,P}(R^{\nu +83/64+\epsilon })$$

arXiv comment34 pages, 1 table