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

Abstract

Let $x⩾2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $\psi (x;q,a):={\sum }_{n⩽x,n\equiv a(\mathrm{mod}q)}\Lambda (n)$, 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 $q⩽x^{1/2+1/231}$. 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 $q$.

To appear in J. Lond. Math. Soc.; 34 pages; Minor corrections and changes have been applied to the first version