On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas

Igor E. Shparlinski¹

¹ Department of Computing Macquarie University North Ryde, NSW 2109, Australia

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) no. 1, pp. 1-7

Résumé

\def\substack#1\\#2*{{\textstyle{#1\atop#2}}}\def\mand{\quad\mbox{and}\quad} \def\cH{{\cal H}} For a prime $p>2$, an integer $a$ with $\gcd(a,p)=1$ and real $1\le X,Y p$, we consider the set of points on the modular hyperbola $$ \mathcal H_{a,p}(X,Y) = \{(x,y) : xy\equiv a\pmod p,\, 1 \le x\le X,\, 1 \le y\le Y\}. $$ We give asymptotic formulas for the average values $$ \sum_{\substack (x,y)\in \mathcal H_{a,p}(X,Y)\\ x \ne y*}\frac{\varphi(|x-y|)}{|x-y|}\quad\text{and}\quad \sum_{\substack (x,y)\in \mathcal H_{a,p}(X,X)\\ x \ne y*} \varphi(|x-y|) $$ with the Euler function $\varphi(k)$ on the differences between the components of points of $\mathcal H_{a,p}(X,Y)$.

Détail

DOI : 10.4064/ba56-1-1

Keywords: def substack * textstyle atop def mand quad mbox quad def cal prime integer gcd real consider set points modular hyperbola mathcal equiv pmod asymptotic formulas average values sum substack mathcal y* frac varphi x y x y quad text quad sum substack mathcal y* varphi x y euler function varphi differences between components points mathcal