Visible points on multidimensional modular hyperbolas

For integers q⩾1, s⩾3 and a with gcd(a,q)=1 and a real U⩾0, we obtain an asymptotic formula for the number of integer points (u1,…,us)∈s[1,U] on the s-dimensional modular hyperbola 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.