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

Suppose $a,b$ are fixed algebraic numbers with $1\le a<b$. Let ${\mathrm{\Delta }}_{a,b}(x)$ be the error term for the number of lattice points in a two-dimensional area $h^a{r}^b\le x$ with $h,r$ positive integers. In this paper, we establish an asymptotic formula for the mean square of ${\mathrm{\Delta }}_{a,b}(x)$ when $a,b$ are fixed algebraic numbers such that ${\displaystyle \frac{a}{b}}$ 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 ${\mathrm{\Delta }}_{a,b}(x)$.