On Goldbach numbers in short intervals

TL;DR

Assuming the Riemann Hypothesis, the paper proves that every sufficiently large number has a nearby even number that is a sum of two primes within a short interval, improving previous bounds.

Contribution

The paper improves the constant in the short interval for Goldbach numbers under the Riemann Hypothesis from 9696 to 123.

Findings

Existence of Goldbach numbers in intervals of length approximately 123 log^2 x

Enhanced bounds compared to previous work by Cully-Hugill and Dudek

Conditional proof assuming the Riemann Hypothesis

Abstract

Assuming the Riemann Hypothesis, we prove that for all $x\geq 2$, there exists at least one even integer within the interval $(x, x+123\log^2x]$, that can be expressed as the sum of two primes. This result is an improvement over the recent work of Cully-Hugill and Dudek, who obtained the constant $9696$ instead of $123$.