2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228010This paper expands and improves on the general Sieve method. This expaned and improved Sieve is applied to Goldbach's problem. A new estimate of the exception set in Goldbach's number E(X), an improved lower bound D_{1,2}(N) and upper bound D(N) are proposed. The proposed values are: |E(X)|<= X^{0.702+e}, D_{1,2}(N)>= 2.27C(N)/ln^{2}(N), D(N) <= 6.916C(N)/ln^{2}(N).50 pages, 5 tablesNumber Theory11A99, 11Y99text