Prime Distribution and Siegel Zeroes

Abstract

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. For a fixed $R>7$, we prove that under the assumption of an exceptional character $\chi$ with $L(1,\chi)<(\log D)^{-R}$, there exists a range of $x$ for which the asymptotic $$\psi(x,q,a)=\frac{\psi(x)}{\phi(q)}\left(1-\chi\left(\frac{aD}{(q,D)}\right)+o(1)\right)$$ holds for $q<x^{\frac{30}{59}-\varepsilon}$. 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 $Q<x^{\frac{16}{31}-\varepsilon}$. This improves on a Friedlander and Iwaniec 2003 result that requires $q<x^{\frac{233}{462}}$ and $R\geq 554,401^{554,401}$.