On Goldbach numbers in short intervals

Abstract

Assuming the Riemann Hypothesis, we prove that for all $x\ge 2$, there exists at least one even integer within the interval $(x,x+123{\log }^{2}x]$, 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 $123$.