For positive integers m, U and V, we obtain an asymptotic formula for the number of integer points (u,v) ∈ [1,U] × [1,V] which belong to the modular hyperbola uv ≡ 1 (mod m) and also have gcd(u,v) =1, which are also known as primitive points. Such points have a nice geometric interpretation as points on the modular hyperbola which are "visible" from the origin.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-3-1,
author = {Igor E. Shparlinski},
title = {Primitive Points on a Modular Hyperbola},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {54},
year = {2006},
pages = {193-200},
zbl = {1153.11322},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-3-1}
}
Igor E. Shparlinski. Primitive Points on a Modular Hyperbola. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 54 (2006) pp. 193-200. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba54-3-1/