On the mean square of the error term for the number of lattice points in a two-dimensional area

Abstract

Suppose $a,~b$ are fixed algebraic numbers with $1\leq a<b$. Let $\Delta_{a,b}(x)$ 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 $\Delta_{a,b}(x)$ 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 $\Delta_{a,b}(x)$.