Primes in arithmetic progressions and Siegel zeroes

Source abstract

Let $χ$ be a Dirichlet character mod $D$ with $L(s,χ)$ its associated $L$-function, and let $ψ(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. Let $χ$ be a primitive character modulo $D$, and let $ν>0$ be small. We prove that if $L(s,χ)$ has a Siegel zero at $s=β=1-\frac{1}{η\log D}$ with $η>η_0(ν)$ for some large $η_0(ν)$, there exists a range of $x$ for which the asymptotic $$ψ(x,q,a)=\frac{ψ(x)}{φ(q)}\left[1+O(\varepsilon_{η_0})-χ\left(\frac{aD}{(q,D)}\right)\right]$$ holds for $q<x^{\frac{30}{59}-ν}$. We also show slightly better bounds for $q$ if we take an average over a range of $q$, finding an Elliott-Halberstam-type result for $q\sim Q$ on the range $Q<x^{\frac{16}{31}-ν}$. This improves on a 2003 result of Friedlander and Iwaniec that requires $q<x^{\frac{233}{462}}$ and builds on recent work of Sachpazis.