Linnik's large sieve and the $L^{1}$ norm of exponential sums

TL;DR

This paper explores the behavior of the $L^{1}$ norm of exponential sums over square-free integers and primes using Linnik's large sieve, providing new proofs and insights into classical bounds.

Contribution

It introduces a novel approach combining Balog-Ruzsa and Linnik's large sieve to analyze exponential sums and offers a new proof of Vaughan's lower bound involving the von Mangoldt function.

Findings

Established bounds for $L^{1}$ norms of exponential sums over primes and square-free integers.

Provided a new proof of Vaughan's lower bound for exponential sums with the von Mangoldt function.

Connected Ramanujan's sum with the analysis of exponential sums.

Abstract

The method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the lower bound due to Vaughan for the $L^{1}$ norm of an exponential sum with the von Mangoldt $\Lambda$ function over the primes is furnished. Ramanujan's sum arises naturally in the proof, which also employs Linnik's large sieve.