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

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.