The large sieve with power moduli for $\mathbb{Z}[i]$

TL;DR

This paper extends the large sieve inequality to power moduli in the Gaussian integers, using advanced harmonic analysis techniques to generalize classical results from integers to complex quadratic integers.

Contribution

It introduces a new large sieve inequality for power moduli in 1, building upon previous work for integers and employing harmonic analysis methods.

Findings

Established a large sieve inequality for 1 with power moduli.

Extended classical large sieve results from integers to Gaussian integers.

Utilized Weyl differencing and Poisson summation in the proof.

Abstract

We establish a large sieve inequality for power moduli in $\mathbb{Z}[i]$, extending earlier work by L. Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in $\mathbb{Z}$. Our method starts with a version of the large sieve for $\mathbb{R}^2$. We convert the resulting counting problem back into one for $\mathbb{Z}[i]$ which we then attack using Weyl differencing and Poisson summation.