Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds
Abstract
For n≥2, 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 𝒞n 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 lvert𝒞nrvert/A(n)≥ (e−γ−o(1))log A(n)/loglog 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 (loglog 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 Nk and Mk be the least integers such that ϕ(n)>kπ(n) for every n≥ Nk and B(n)>kA(n) for every n≥ Mk, respectively. We prove Mk≤ Nk+1 for every k≥1, determine N1,…,N6 and M1,…,M5 exactly, and obtain Mk=Nk+1 for 1≤ k≤5.
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BibTeX
@misc{fatehizadeh2026prime,
title = {{Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds}},
author = {Amirali Fatehizadeh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13852},
eprint = {2609.13852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13852},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds
AU - Amirali Fatehizadeh
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13852
UR - https://arxiv.org/abs/2609.13852
AB - For n≥2, 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 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 lvert𝒞_nrvert/A(n)≥ (e^−γ−o(1))log A(n)/loglog 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 (loglog 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≥ N_k and B(n)>kA(n) for every n≥ M_k, respectively. We prove M_k≤ N_k+1 for every k≥1, determine N_1,…,N_6 and M_1,…,M_5 exactly, and obtain M_k=N_k+1 for 1≤ k≤5.
ER -