On existence of generalised Lehmer numbers modulo primes

In this article, we discuss a multivariable analogue of Lehmer’s parity problem. We generalise the notions of Lehmer numbers as Lehmer $q$-tuples for a fixed odd prime $q$. We count the number of Lehmer $q$-tuples and derive an asymptotic estimate with explicit error terms. This deals with estimating various character sums and Deligne’s bound for incomplete exponential sums.