关于整除关系σ(n)｜σ(n+h)及广义Erdős–Sierpiński猜想

中文摘要 AI 辅助

对于每个固定正整数h，我们研究整除关系σ(n)｜σ(n+h)。我们分离出由移位丰度指数的整数商产生的显式正则族，并证明其补集满足次指数节省；特别地，不超过x的解的数量为Oₕ(x/(log x)²)。我们还研究比例方程σ(n+h)=λσ(n)。对于每个固定非零整数h，对所有实λ>0一致，不超过x的解的数量为O(x/√(log log log x))，当x超过依赖于h的阈值时具有绝对隐含常数。最后，我们给出一个显式族，在Schinzel假设H下，可产生σ(n+1)=2σ(n)的无限多解；Bateman–Horn猜想给出该族中不超过x的成员数量的精确渐近。我们猜想，对于每个固定h,k≥1，σ(n+h)=kσ(n)具有无限多正整数解。

英文摘要

For each fixed positive integer $h$, we study the divisibility relation $σ(n)\midσ(n+h)$. We isolate an explicit regular family arising from integral quotients of shifted abundancy indices and show that the complementary set satisfies a subexponential saving; in particular, the number of solutions up to $x$ is $O_h(x/(\log x)^2)$. We also study the proportionality equation $σ(n+h)=λσ(n)$. For every fixed nonzero integer $h$, uniformly for all real $λ>0$, the number of solutions up to $x$ is $O(x/\sqrt{\log\log\log x})$, with an absolute implied constant once $x$ exceeds an $h$-dependent threshold. Finally, we give an explicit family which, under Schinzel's Hypothesis $H$, produces infinitely many solutions of $σ(n+1)=2σ(n)$; the Bateman--Horn conjecture yields a precise asymptotic for the number of members of this family up to $x$. We conjecture that $σ(n+h)=kσ(n)$ has infinitely many positive integer solutions for every fixed $h,k\ge1$.

发表机构

• Shahid Beheshti University（沙希德·贝赫什蒂大学）
• Stellenbosch University（斯坦陵布什大学）
• University of Oxford（牛津大学）

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