Lattice points under polynomial curves

Abstract: This presentation is concerned with various analogs of the Gauss circle and Dirichlet divisor problems. For a given polynomial $f(x_1,\ldots,x_s)$ with integer coefficients, let $r_f(n) = \#\{(x_1,\ldots,x_s) \in \mathbb{N}^s: f(x_1,\ldots,x_s) = n\}$. Using van der Corput's method to control error terms, we obtain asymptotic formulas for averages of this function of the form $\sum_{n \leq N} r_f(n)$ with various choices of $f$. When $f(x_1,x_2)$ is a polynomial in two variables, this counts the lattice points in the first quadrant bounded by the curve $f(x_1,x_2) = N$. In this setting, we demonstrate how to obtain asymptotics for various curves of low degree as well as how the method can be extended to apply to some simple hypersurfaces bounding points in $\mathbb{N}^s$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)