  In this paper, we obtain sharp estimates for the number of lattice points
under and near the dilation of a general parabola, the former generalizing an
old result of Popov. We apply Vaaler's lemma and the Erd\H{o}s-Turan inequality
to reduce the two underlying counting problems to mean values of a certain
quadratic exponential sums, whose treatment is subject to classical analytic
techniques.
