On Lehmer's problem and related problems
Number Theory
2023-09-15 v3
Abstract
We show that if with composite, then and , together with similar results for the unitary totient function, Dedekind function, and the sum of unitary divisors.
Keywords
Cite
@article{arxiv.2303.16853,
title = {On Lehmer's problem and related problems},
author = {Tomohiro Yamada},
journal= {arXiv preprint arXiv:2303.16853},
year = {2023}
}
Comments
14 pages (Added arguments in cases $N$, $N_1$, or $\omega(N)$ is small. Our results make no sense when $N_1\leq 17$ or $\omega(N)\leq 2$. Moreover, our argument does not work when $N_1=21$ in Theorems 1 or $\omega(N)=3$ in Theorems 2 and 4. Indeed, $N=15, 255$ satisfy $N+1=2\varphi(N)$ but $\log\log\log 15$ and $\log\log\omega(15)$ are negative and $2>21\log\log\omega(255)$)