Siegel Zeros and the Hardy-Littlewood Conjecture

Abstract: 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.