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

Source abstract

Abstract Let and assume that and are coprime positive integers. As usual, , where 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 beyond the square‐root barrier of the generalized Riemann hypothesis. In particular, their asymptotic yields nontrivial information for moduli . 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 in a slightly wider range of .