The exceptional set of Goldbach numbers (II)

1. Introduction. A positive number which is a sum of two odd primes is called a Goldbach number. Let E(x) denote the number of even numbers not exceeding x which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that E(x) = 2 for every x ≥ 4. E(x) is usually called the exceptional set of Goldbach numbers. In [8] H. L. Montgomery and R. C. Vaughan proved that ${\displaystyle E\left(x\right)=O\left(x^{1-\Delta }\right)}$ for some positive constant Δ > 0${\displaystyle .In\left[3\right]Chen\text{and}Panprovedt\hat{o}\ne cantake\Delta >0.01.In\left[6\right],weprovedt\hat{E}\left(x\right)=O\left(x^{0.921}\right)}$. In this paper we prove the following result. Theorem. For sufficiently large x, ${\displaystyle E\left(x\right)=O\left(x^{0.914}\right)}$. Throughout this paper, ε always denotes a sufficiently small positive number that may be different at each occurrence. A is assumed to be sufficiently large, A < Y, and ${\displaystyle D=Y^{1+\epsilon }}$.