Expansion and Improvement of Sieve and application in Goldbach's problem

Abstract

This 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).