Amplification arguments for large sieve inequalities

TL;DR

This paper introduces a novel proof technique for the large sieve inequality using amplification, extending it to new sieve inequalities for modular forms and Hecke eigenvalues.

Contribution

It provides a new proof approach for the arithmetic large sieve inequality and adapts it to modular forms, linking harmonic and arithmetic perspectives.

Findings

New proof of the arithmetic large sieve inequality

Extension of sieve inequalities to modular forms

Connection between Hecke eigenvalues and reductions modulo primes

Abstract

We present a new proof of the "arithmetic" large sieve inequality, starting from the corresponding "harmonic" inequality, which is based on an amplification idea. We show that this also adapts to give some new sieve inequality for modular forms, where Hecke eigenvalues are thought as the analogues of the reductions of integers modulo primes.