Averages over the Gaussian Primes: Goldbach's Conjecture and Improving Estimates

TL;DR

This paper proves versions of Goldbach's conjecture for Gaussian primes within sectors, demonstrating that sufficiently large odd integers can be expressed as sums of three Gaussian primes with arguments in a specified sector.

Contribution

It establishes Goldbach-type results for Gaussian primes in arbitrary sectors and introduces density results for the binary Goldbach conjecture in sectors.

Findings

Every large odd integer in certain conditions is a sum of three Gaussian primes in a sector.

A density version of the binary Goldbach conjecture in sectors is proved.

Provides improved estimates related to Gaussian primes and Goldbach's conjecture.

Abstract

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus \omega ) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in \omega $, for $j=1,2,3$. A density version of the binary Goldbach conjecture in a sector is also proved.