Three-dimensional exponential sums under constant perturbation

In this paper, by generalizing Bombieri and Iwaniec’s double large sieve inequality, we obtain an estimation on a type of three-dimensional exponential sums with constant perturbation. As an application of our estimation, we obtain an asymptotic formula for a sum involving the Mangoldt function and the integral part function. This type of sum was originally studied by Bordelles-Dai-Heyman-Pan-Shparlinski.