Regular integers modulo n

zbMATH Open1199.11026arXiv0710.1936MaRDI QIDQ3539393

László Tóth

Publication date: 18 November 2008

Abstract: Let $n=p_1^{u_1}...p_r^{u_r}>1$ be an integer. An integer $a$ is called regular (mod $n$) if there is an integer $x$ such that $a^{2}xequiva$ (mod $n$). Let $varrho(n)$ denote the number of regular integers $a$ (mod $n$) such that $1lealen$. Here $varrho(n)=(phi(p_1^{u_1})+1)...(phi(p_r^{u_r})+1)$, where $phi(n)$ is the Euler function. In this paper we first summarize some basic properties of regular integers (mod $n$). Then in order to compare the rates of growth of the functions $varrho(n)$ and $phi(n)$ we investigate the average orders and the extremal orders of the functions $varrho(n)/phi(n)$, $phi(n)/varrho(n)$ and $1/varrho(n)$.

Full work available at URL: https://arxiv.org/abs/0710.1936

zbMATH Keywords

multiplicative functionaverageextremal order

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Arithmetic functions; related numbers; inversion formulas (11A25)

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