Lattice Points Problems about the Paraboloid

Abstract

The Gauss circle problem, which asks for the best possible error term when approximating the number of lattice points inside a dilating circle centered at the origin by its area, is a longstanding open question in number theory. One may as well ask similar questions for regions bounded by other conics such as hyperbola and parabola, or their higher dimensional generalizations. Building off of the techniques of Huang and Li, we establish in this thesis asymptotic formulae for the number of lattice points under and near the standard paraboloid of dimension two and higher. The upper bound estimates we obtain on the error terms nearly meet those in the omega result of Chamizo and Pastor, and therefore are essentially best possible.