Visible points on multidimensional modular hyperbolas Original Research Article

For integers qgreater-or-equal, slanted1, sgreater-or-equal, slanted3 and a with gcd(a,q)=1 and a real Ugreater-or-equal, slanted0, we obtain an asymptotic formula for the number of integer points (u1,…,us)set membership, variant[1,U]s on the s-dimensional modular hyperbola image with the additional property gcd(u1,…,us)=1. Such points have a geometric interpretation as points on the modular hyperbola which are “visible” from the origin. This formula complements earlier results of the first author for the case s=2 and a=1. Moreover, we prove stronger results for smaller U on “average” over all a. The proofs are based on the Burgess bound for short character sums.