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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.14176 |
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| _version_ | 1866913743629189120 |
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| author | Jia, Lirui Zhai, Wenguang |
| author_facet | Jia, Lirui Zhai, Wenguang |
| contents | 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 $Δ_{a,b}(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the mean square of the error term for the number of lattice points in a two-dimensional area Jia, Lirui Zhai, Wenguang Number Theory 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 $Δ_{a,b}(x)$. |
| title | On the mean square of the error term for the number of lattice points in a two-dimensional area |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.14176 |