发表机构
University of Northern British Columbia; University of Lethbridge（不列颠哥伦比亚大学; 莱思布里奇大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对特定参数范围的广义移位三元加性除数和，在假设相关函数算术级数分布水平的条件下，证明了带显式主项和幂节省误差项的渐近公式。
AI 中文摘要
本文证明了一个条件性结果，该结果给出了广义移位三元加性除数和$D_{\mathcal{I},\mathcal{J}}(X,1)=\sum_{n \leq X} \tau_{\mathcal{I}}(n) \tau_{\mathcal{J}}(n + 1)$的渐近公式，其中$\mathcal{I}=\{\alpha_1,\alpha_2,\alpha_3\}$，$\mathcal{J} = \{\beta_1, \beta_2, \beta_3\}$，且对$i=1,2,3$有$|\alpha_i|, |\beta_i|\leq \delta$，$0<\delta<1/44$。主定理的证明依赖于假设广义三元加性除数函数$\tau_{\mathcal{J}}$在算术级数中直至水平$2/3$的平均分布水平。
英文摘要
In this paper, we prove a conditional result that yields an asymptotic formula with an explicit main term and a power-saving error term for the generalized shifted ternary additive divisor sum $D_{\mathcal{I},\mathcal{J}}(X,1)=\sum_{n \leq X} τ_{\mathcal{I}}(n) τ_{\mathcal{J}}(n + 1)$, where $\mathcal{I}=\{α_1,α_2,α_3\}$ and $\mathcal{J} = \{β_1, β_2, β_3\}$, with $|α_i|, |β_i|\leq δ$ for $i=1,2,3$, and $0<δ<1/44$. The proof of our main theorem relies on assuming an averaged level of distribution for the generalized ternary additive divisor function $τ_{\mathcal{J}}$ in arithmetic progressions up to the level $2/3$.
Comments30 pages
