Goldbach Numbers in Short Intervals -- A Nonnegative Model Approach

TL;DR

This paper improves bounds on the interval length needed to ensure most even numbers are sums of two primes by using a novel nonnegative model approach with the Circle Method, avoiding complex inequalities.

Contribution

It introduces a new nonnegative model combined with the Circle Method to reduce the interval length for Goldbach representations without relying on vector sieve inequalities.

Findings

Reduced the shortest interval length for Goldbach representations.

Achieved results without using vector sieve inequalities.

Improved upon previous bounds established by Harman.

Abstract

We decrease the length of the shortest interval for which almost all even integers in it are the sum of two primes. This is achieved by applying a version of the Circle Method that uses two minorants together with a nonnegative model for one of them. Compared to Harman's previous strongest result of this type, we in this way do not need any vector sieve type inequality and so require neither additional majorants nor strong density requirements for the minorants.