Three-dimensional exponential sums under constant perturbation

TL;DR

This paper generalizes a large sieve inequality to estimate three-dimensional exponential sums with constant perturbation and applies this to derive an asymptotic formula for a sum involving the Mangoldt function.

Contribution

It introduces a generalized large sieve inequality for three-dimensional exponential sums with constant perturbation, extending prior methods.

Findings

Derived an asymptotic formula for a sum involving the Mangoldt function.

Provided bounds for three-dimensional exponential sums with perturbation.

Extended the applicability of large sieve techniques.

Abstract

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.