# On two lattice points problems about the parabola

## TL;DR

This paper derives optimal asymptotic formulas with minimal error terms for counting lattice points under and near a dilated parabola, advancing understanding in this specific geometric number theory problem.

## Contribution

It provides the first optimal error bounds for lattice points near the parabola and improves previous results for points under the parabola, using advanced Fourier and number theory techniques.

## Key findings

- Optimal asymptotic formulas with error terms for lattice points under the parabola
- Sharp upper bounds for lattice points near the parabola
- Achievement of square root cancellation in the parabola context

## Abstract

We obtain asymptotic formulae with optimal error terms for the number of lattice points under and near a dilation of the standard parabola, the former improving upon an old result of Popov. These results can be regarded as achieving the square root cancellation in the context of the parabola, whereas its analogues are wide open conjectures for the circle and the hyperbola. We also obtain essentially sharp upper bounds for the latter lattice points problem. Our proofs utilize techniques in Fourier analysis, quadratic Gauss sums and character sums.