The exceptional set of Goldbach problem and Linnik's constant

TL;DR

This paper establishes an upper bound on the number of even integers not expressible as the sum of two primes and provides bounds related to the least prime in arithmetic progressions, advancing understanding of Goldbach's conjecture.

Contribution

It introduces new bounds for the exceptional set in Goldbach's problem and for the least prime in arithmetic progressions, using novel analytic techniques.

Findings

E(X)=O(X^{7/10}) for the exceptional set

P(q)=O(q^5) for the least prime in progressions

Ineffective constants in bounds

Abstract

Let $E(X)$ denote the number of even integers below $X$ which are not a sum of two primes. We prove the bound $E(X)=O(X^{\frac{7}{10}})$, where the implicit constant is ineffective. The method applied here also leads to $P(q)=O(q^5)$, where $P(q)$ denotes the least prime, if it exists, in any arithmetic progression modulo $q$.