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

Igor E. Shparlinski. "On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas." Bulletin of the Polish Academy of Sciences. Mathematics 56.1 (2008): 1-7. <http://eudml.org/doc/281163>.

@article{IgorE2008,
abstract = {For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola $_\{a,p\}(X,Y) = \{(x,y) : xy ≡ a(mod p), 1 ≤x≤X, 1 ≤y≤Y\}$. We give asymptotic formulas for the average values $∑_\{\begin\{array\}\{c\}(\end\{array\}x,y)∈ _\{a,p\}(X,Y) x ≠ y*\} φ(|x-y|)/|x-y|$ and $∑_\{\begin\{array\}\{c\}(\end\{array\}x,y)∈ _\{a,p\}(X,X) x ≠ y*\} φ(|x-y|)$ with the Euler function φ(k) on the differences between the components of points of $_\{a,p\}(X,Y)$.},
author = {Igor E. Shparlinski},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {modular hyperbola; Euler function; discrepancy},
language = {eng},
number = {1},
pages = {1-7},
title = {On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas},
url = {http://eudml.org/doc/281163},
volume = {56},
year = {2008},
}

TY - JOUR
AU - Igor E. Shparlinski
TI - On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2008
VL - 56
IS - 1
SP - 1
EP - 7
AB - For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola $_{a,p}(X,Y) = {(x,y) : xy ≡ a(mod p), 1 ≤x≤X, 1 ≤y≤Y}$. We give asymptotic formulas for the average values $∑_{\begin{array}{c}(\end{array}x,y)∈ _{a,p}(X,Y) x ≠ y*} φ(|x-y|)/|x-y|$ and $∑_{\begin{array}{c}(\end{array}x,y)∈ _{a,p}(X,X) x ≠ y*} φ(|x-y|)$ with the Euler function φ(k) on the differences between the components of points of $_{a,p}(X,Y)$.
LA - eng
KW - modular hyperbola; Euler function; discrepancy
UR - http://eudml.org/doc/281163
ER -