Linnik's theorem on Goldbach numbers in short intervals

Abstract

The Goldbach conjecture states that every even number larger than 2 can be written as the sum of two primes. We shall therefore call an even number a Goldbach number if it can be written as the sum of two primes in at least one way. It has been known for a long time that almost all even numbers are Goldbach numbers. In fact, Montgomery and Vaughan [14] have shown that if E(N) denotes the number of even numbers less than or equal to N which are not Goldbach numbers, then there exists an absolute constant δ>0 such that