## A new explicit formula in the additive theory of primes with applications I. The explicit formula for the Goldbach problem and the Generalized Twin Prime Problem

### Volume 210 / 2023

#### Abstract

We prove an approximative formula for the contribution of the major arcs for the Goldbach decomposition of even numbers. The formula does not necessarily give an asymptotic but it expresses the contribution of the major arcs with the aid of a bounded number of possible “generalized exceptional zeros” of Dirichlet $L$-functions lying near the boundary line Re$s = 1$. The advantage of the formula over earlier ones is that it makes it possible to define a much larger set as major arcs. It can be considered as a generalization of a method of Montgomery and Vaughan (1975) in their breakthrough paper about estimating the size of the exceptional set in Goldbach’s problem. Our method has many applications for possibly existing exceptional Goldbach numbers (which cannot be written as a sum of two primes).