On the distribution of points on multidimensional modular hyperbolas

Abstract

We study the distribution of points on the $(n+1)$-dimensional modular hyperbola $a_1\cdots a_{n+1} \equiv c \pmod q$, where $q$ and $c$ are relatively prime integers. In particular, we show that an elementary argument leads to a straight-forward proof of a recent result of T.~Zhang and W.~Zhang, with a stronger error term. We also use character sums to obtain an asymptotic formula for the number of points in a given box that lie on such hyperbolas.