A modified Gallagher's Lemma

TL;DR

This paper introduces a modified version of Gallagher's Lemma using Cesaro weights, and applies it to derive new mean value estimates for Dirichlet polynomials, enhancing analytical tools in number theory.

Contribution

The paper presents a novel modification of Gallagher's Lemma with Cesaro weights and demonstrates its application to Dirichlet polynomial mean value estimates.

Findings

Modified Gallagher's Lemma with Cesaro weights proved.

New mean value estimates for Dirichlet polynomials established.

Abstract

First we prove a modified version of the famous Lemma on the mean square estimate for exponential sums, by plugging the Cesaro weights in the right hand side of Gallagher's inequality. Then we apply it, in order to establish a mean value estimate for the Dirichlet polynomials.