On Goldbach numbers in short intervals

Abstract

Assuming the Riemann Hypothesis, we prove that for all $$x\geqslant 2$$ , 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.