素数与非素数互素剩余：一个尖锐构造与精确阈值
Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds
- Faculty of Mathematical Sciences, Shahid Beheshti University（沙希德·贝赫什提大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了合数互素剩余族的显式单射参数化，证明其与素数互素剩余数之比达到最优常数e^{-γ}，并精确确定了若干阈值N_k与M_k及其相等关系。
AI中文摘要：
对于\\(n\ge2\\)，设\\(A(n)=\pi(n)-\omega(n)\\)和\\(B(n)=\phi(n)-\pi(n)+\omega(n)\\)分别表示\\(n\\)的素数与非素数互素剩余（totative）的个数，其中\\(1\\)计入\\(B(n)\\)。我们通过将适当受限的素数互素剩余的无平方因子乘积与一个作为唯一最大素因子的较大素数互素剩余相结合，构造了一个显式的单射参数化族\\(\mathcal C_n\\)的合数互素剩余，使得该参数化可恢复。我们证明\\(\lvert\mathcal C_n\rvert/A(n)\ge (e^{-\gamma}-o(1))\log A(n)/\log\log A(n)\\)，实现了尖锐的经典首项常数\\(e^{-\gamma}\\)。对于同一族，使用相同的定义参数，我们获得了一个有效改进，误差阶为\\((\log\log A(n))^{-1/4}\\)，且常数为绝对有效可计算的。经典最小阶结果和素数定理表明\\(e^{-\gamma}\\)是最优的：包含在非素数互素剩余中的任何族都不能在此尺度上以更大的首项常数满足一致下界。我们还确定了精确的最终阈值。设\\(N_k\\)和\\(M_k\\)分别为使得对所有\\(n\ge N_k\\)有\\(\phi(n)>k\pi(n)\\)和对所有\\(n\ge M_k\\)有\\(B(n)>kA(n)\\)的最小整数。我们证明对每个\\(k\ge1\\)有\\(M_k\le N_{k+1}\\)，精确确定了\\(N_1,\ldots,N_6\\)和\\(M_1,\ldots,M_5\\)，并得到对\\(1\le k\le5\\)有\\(M_k=N_{k+1}\\)。
英文摘要：
For \(n\ge2\), let \(A(n)=π(n)-ω(n)\) and \(B(n)=ϕ(n)-π(n)+ω(n)\) denote the numbers of prime and nonprime totatives of \(n\), respectively, with \(1\) included in \(B(n)\). We construct an explicit injectively parametrized family \(\mathcal C_n\) of composite totatives by completing suitably restricted squarefree products of prime totatives with a larger prime totative that is the unique largest prime factor, making the parametrization recoverable. We prove \(\lvert\mathcal C_n\rvert/A(n)\ge (e^{-γ}-o(1))\log A(n)/\log\log A(n)\), realizing the sharp classical leading constant \(e^{-γ}\). For the same family, with the same defining parameters, we obtain an effective refinement with error of order \((\log\log A(n))^{-1/4}\) and absolute effectively computable constants. Classical minimal-order results and the prime number theorem show that \(e^{-γ}\) is optimal: no family contained in the nonprime totatives can satisfy a uniform lower bound at this scale with a larger leading constant. We also determine exact eventual thresholds. Let \(N_k\) and \(M_k\) be the least integers such that \(ϕ(n)>kπ(n)\) for every \(n\ge N_k\) and \(B(n)>kA(n)\) for every \(n\ge M_k\), respectively. We prove \(M_k\le N_{k+1}\) for every \(k\ge1\), determine \(N_1,\ldots,N_6\) and \(M_1,\ldots,M_5\) exactly, and obtain \(M_k=N_{k+1}\) for \(1\le k\le5\).
