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

Abstract

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