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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.07121 |
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| _version_ | 1866918193859133440 |
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| author | Guo, Zhen Li, Jinjiang Long, Linji Zhang, Min |
| author_facet | Guo, Zhen Li, Jinjiang Long, Linji Zhang, Min |
| contents | Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in\mathbb{Z}^+$, denote by $τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$ the asymmetric two--dimensional divisor function with congruence conditions, i.e., \begin{equation*} τ_{a,b}(n;\ell_1,M_1,l_2,M_2)=\sum_{\substack{n=n_1^an_2^b\\ n_1\equiv\ell_1\!\!\!\!\!\pmod{M_1}\\ n_2\equiv\ell_2\!\!\!\!\!\pmod{M_2}}}1. \end{equation*} In this paper, we shall establish an asymptotic formula of the mean square of the error term of the sum $\sum_{n\leqslant M_1^aM_2^bx}τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$. This result constitutes an enhancement upon the previous result of Zhai and Cao [16]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the mean square of the error term for the asymmetric two-dimensional divisor problem with congruence conditions Guo, Zhen Li, Jinjiang Long, Linji Zhang, Min Number Theory Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in\mathbb{Z}^+$, denote by $τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$ the asymmetric two--dimensional divisor function with congruence conditions, i.e., \begin{equation*} τ_{a,b}(n;\ell_1,M_1,l_2,M_2)=\sum_{\substack{n=n_1^an_2^b\\ n_1\equiv\ell_1\!\!\!\!\!\pmod{M_1}\\ n_2\equiv\ell_2\!\!\!\!\!\pmod{M_2}}}1. \end{equation*} In this paper, we shall establish an asymptotic formula of the mean square of the error term of the sum $\sum_{n\leqslant M_1^aM_2^bx}τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$. This result constitutes an enhancement upon the previous result of Zhai and Cao [16]. |
| title | On the mean square of the error term for the asymmetric two-dimensional divisor problem with congruence conditions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.07121 |