Regular integers modulo n

Abstract

Let $n=p_1^{{\nu }_1}...p_r^{{\nu }_r}>1$ be an integer. An integer $a$ is called regular (mod $n$) if there is an integer $x$ such that $a^{2}x\equiv a$ (mod $n$). Let $\varrho (n)$ denote the number of regular integers $a$ (mod $n$) such that $1\le a\le n$. Here $\varrho (n)=(\varphi (p_1^{{\nu }_1})+1)...(\varphi (p_r^{{\nu }_r})+1)$, where $\varphi (n)$ 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 $\varphi (n)$ we investigate the average orders and the extremal orders of the functions $\varrho (n)/\varphi (n)$, $\varphi (n)/\varrho (n)$ and $1/\varrho (n)$.

9 pages, final version