Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes

Let χ be a Dirichlet character mod D with L (s, χ) its associated L-function, and let ψ (x, q, a) be Chebyshev's prime-counting function for primes congruent to a modulo q. We show that under the assumption of an exceptional character χ with L (1, χ) =o ( (D) ^-5), for any q<x^ 23-, the asymptotic ψ (x, q, a) =ψ (x) ϕ (q) (1-χ (aD (D, q) ) +o (1) ) holds for almost all a with (a, q) =1. We also find that for any fixed a, the above holds for almost all q<x^ 23- with (a, q) =1. Previous prime equidistribution results under the assumption of Siegel zeroes (by Friedlander-Iwaniec and the current author) have found that the above asymptotic holds either for all a and q or on average over a range of q (i. e. for the Elliott-Halberstam conjecture), but only under the assumption that q<x^θ where θ=3059 or 1631, respectively.