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 ω⊂𝕋. There is an integer N_ω, so that every odd integer n with N(n)>N_ω and dist( arg(n) , 𝕋∖ω ) > (log N(n)) ^-B, is a sum of three Gaussian primes n=p_1+p_2+p_3, with arg(p_j) ∈ω, for j=1,2,3. A density version of the binary Goldbach conjecture in a sector is also proved.