  In 2016, Fei \cite{fei2016application} established a bound on the Siegel
zeros for real primitive Dirichlet characters modulo $q$, assuming the weak
Hardy-Littlewood conjecture. Building on Fei's work, Jia
\cite{jia2022conditional} demonstrated the same bound using a stronger version
of the Hardy-Littlewood conjecture. In this paper, we present a slightly
simplified approach to reprove their results.
