The Large Sieve Inequality for Integer Polynomial Amplitudes

TL;DR

This paper improves the large sieve inequality for sequences weighted by polynomial values with integer coefficients of degree at least two, approaching the optimal bounds.

Contribution

It provides a near-optimal version of the large sieve inequality specifically for polynomial amplitude sequences of degree two or higher.

Findings

Achieves a near-best bound for the large sieve with polynomial amplitudes.

Extends the large sieve inequality to polynomial sequences of degree ≥ 2.

Enhances understanding of sieve methods for polynomial-generated sequences.

Abstract

We obtain a close to the best possible version of the large sieve inequality with amplitudes given by the values of a polynomial with integer coefficients of degree $\geq 2$.