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Let $\chi$ be a Dirichlet character mod $D$, let $L(s,\chi)$ be itsassociated $L$-function, and let $\psi(x,q,a)$ denote Chebyshev'sprime-counting function. For a fixed $R>9$, we prove that under the assumptionof an exceptional character $\chi$ with $L(1,\chi)<\log^{-R} D$, there exists arange of $x$ for which$$\psi(x,q,a)=\frac{\psi(x)}{\phi(q)}\left(1-\chi\left(\frac{aD}{(q,D)}\right)+o(1)\right)$$holds for $q<x^{\frac{16}{31}-\epsilon}$, as well as slightly better bounds for$q$ if we take an average over a range of $q$. This improves on a Friedlanderand Iwaniec 2003 result that requires $q<x^{\frac{233}{462}-\epsilon}$ and$R\geq 554,401^{554,401}$.  As a corollary, we prove a Brun-Titchmarsh theorem for primes in congruenceclasses $a$ mod $q$ where $D|q$ and $\chi(a)=1$, i.e. the classes where the setof primes is sparse under the assumption of a Siegel zero.

Prime Distribution and Siegel Zeroes