A Bombieri--Vinogradov Theorem with products of Gaussian primes as moduli

TL;DR

This paper extends the Bombieri--Vinogradov Theorem to moduli formed by products of Gaussian primes expressed as multivariable polynomials, utilizing advanced sieve inequalities and recent progress in Vinogradov's mean value theorem.

Contribution

It adapts Vaughan's proof to Gaussian prime products, incorporating a polynomial large sieve inequality and recent Vinogradov mean value theorem advances.

Findings

Extended range for variables in the theorem

Successful adaptation of Vaughan's proof to Gaussian prime products

Improved bounds using polynomial large sieve inequality

Abstract

We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical Bombieri--Vinogadov Theorem to this setting, we apply the polynomial large sieve inequality that has been recently proved and which includes recent progress in Vinogradov's mean value theorem due to Parsell et al. From the benefit of these improvements, we obtain an extended range for the variables compared to the range obtained from standard arguments only.