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

TL;DR

This paper enhances the sieve method and applies it to Goldbach's problem, providing new bounds and estimates for the exception set and prime representations, advancing understanding of Goldbach's conjecture.

Contribution

The paper introduces an improved sieve technique and derives new bounds for the exception set and prime representations in Goldbach's problem.

Findings

|E(X)| <= X^{0.702+e} for the exception set

D_{1,2}(N) >= 2.27C(N)/ln^{2}(N)

D(N) <= 6.916C(N)/ln^{2}(N)

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).