Showing papers by "László Tóth published in 2007"

Regular integers modulo n

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 $a^2x\equiv a$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers $a$ (mod $n$) such that $1\le a\le n$. Here $\varrho(n)=(\phi(p_1^{ u_1})+1)... (\phi(p_r^{ u_r})+1)$, 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 $1/\varrho(n)$.

An order result for the exponential divisor function

Abstract: The integer $d=\prod_{i=1}^s p_i^{b_i}$ is called an exponential divisor of $n=\prod_{i=1}^s p_i^{a_i}>1$ if $b_i \mid a_i$ for every $i\in \{1,2,...,s\}$. Let $\tau^{(e)}(n)$ denote the number of exponential divisors of $n$, where $\tau^{(e)}(1)=1$ by convention. The aim of the present paper is to establish an asymptotic formula with remainder term for the $r$-th power of the function $\tau^{(e)}$, where $r\ge 1$ is an integer. This improves an earlier result of {\sc M. V. Subbarao} [5].