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

Abstract

Let $x\geqslant 2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $\psi(x;q,a):=\sum_{n\leqslant x,n\equiv a(\!\!\!\mod{\!\!q})}\Lambda(n)$, where $\Lambda$ 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 $q\leqslant x^{1/2+1/231}$. 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 $q$.