A mean value theorem on the binary Goldbach problem and its application

摘要

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p 1 + p 2 = n is solvable in prime variables p 1, p 2 such that p 1 + 2 = P 3, and for every sufficiently large odd integer ${bar n}$ satisfying ${bar n}$ ≢ 1(mod 6), the equation p 1 + p 2 + p 3 = ${bar n}$ is solvable in prime variables p 1, p 2, p 3 such that p 1 + 2 = P 2, p 2 + 2 = P 3. Here P k denotes any integer with no more than k prime factors, counted according to multiplicity.