Primes in arithmetic progressions under the presence of Landau-Siegel zeroes

Let $x⩾2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $\psi (x;q,a):={\sum }_{n⩽x,n\equiv a(\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}q)}\mathrm{\Lambda }(n)$, where $\mathrm{\Lambda }$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of exceptional characters corresponding to “extreme” Landau-Siegel zeroes and established a meaningful asymptotic formula for $\psi (x;q,a)$ beyond the square-root barrier of the Generalized Riemann Hypothesis. In particular, their asymptotic yields non-trivial information for moduli $q⩽x^{1\mathrm{/}2+1\mathrm{/}231}$. In this paper, we considerably relax the extremity of the Landau-Siegel zero required in the work of Friedlander and Iwaniec and obtain a conditional asymptotic formula for $\psi (x;q,a)$ in a slightly wider range of $q$.