Lattice points in elliptic paraboloids

Abstract

We consider the lattice point problem corresponding to a family of elliptic paraboloids in ${\mathbb{R}}^d$ with $d\ge 3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$ because the optimal exponent is conjectural even for the sphere. We also treat some aspects of the case $d=2$, getting for a simple parabolic region an $\Omega$-result that is unknown for the classical circle and divisor problems.

15 pages, added more precise Ω-results