Title: Primes in arithmetic progressions under Siegel zeroes

Abstract: Let x ≥ 1 and let q and a be two coprime positive integers. As usual,

ψ(x;q,a) := sum_{n ≤ x: n = a (mod q)} Λ(n),

where Λ(n) is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of ''extreme'' Siegel zeroes and established an asymptotic formula for ψ(x;q,a) beyond the limitations of GRH, with moduli q beyond √x yielding non-trivial information. In particular, they obtained a meaningful asymptotic for q ≤ x^{1/2+1/231}. We will see how one can relax the ''extremity'' of the exceptional zeroes and replace it by simply the definition of a Siegel zero. We will also discuss an idea to improve the Friedlander-Iwaniec regime and reach the range q ≤ x^{1/2 + 1/82 - ε}. This talk is based on on-going work.