On the mean square of the error term for the number of lattice points in a two-dimensional area
Lirui Jia, Wenguang Zhai
TL;DR
The paper analyzes the mean-square of the error term $Δ_{a,b}(x)$ counting lattice points under $h^a r^b \le x$ for fixed algebraic $1\le a<b$, proving a precise mean-square asymptotic when $a/b$ is irrational and improving the error term for $(a,b)=1$. It develops a Voronoi-type decomposition, isolates a main oscillatory contribution, and controls cross-terms via auxiliary sums and Diophantine bounds, yielding a main term proportional to $\int x^{1/(a+b)}dx$ with a sharp exponentially small error. The authors then translate these mean-square results into quantitative sign-change statements for $Δ_{a,b}(x)$, deriving intervals in which the error term must oscillate. The work extends two-dimensional divisor-type mean-square theory to algebraic $a,b$ and provides explicit constants and convergent series $G_{a,b}$ that govern the asymptotics, with sharper results in the integer-coprime case and implications for the distribution of lattice-point discrepancies.
Abstract
Suppose $a,~b$ are fixed algebraic numbers with $1\leq a<b$. Let $Δ_{a,b}(x)$ be the error term for the number of lattice points in a two-dimensional area $h^ar^b\leq x $ with $h, r$ positive integers. In this paper, we establish an asymptotic formula for the mean square of $Δ_{a,b}(x)$ when $a, b$ are fixed algebraic numbers such that $\dfrac{a}{b}$ is irrational, and improve the error term in the previous asymptotic formula for $a, b$ integers with $(a, b)=1$. Based on these asymptotic formulas, we derive estimates for the sign changes of $Δ_{a,b}(x)$.