Primes in Arithmetic Progressions to Large Moduli 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 Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an exceptional character $\chi$ with $L(1,\chi)=o\left((\log D)^{-5}\right)$, for any $q<x^{\frac 23-\varepsilon}$, the asymptotic $$\psi(x,q,a)=\frac{\psi(x)}{\phi(q)}\left(1-\chi\left(\frac{aD}{(D,q)}\right)+o(1)\right)$$ holds for almost all $a$ with $(a,q)=1$. We also find that for any fixed $a$, the above holds for almost all $q<x^{\frac 23-\varepsilon}$ 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 $q<x^{\theta}$ where $\theta=\frac{30}{59}$ or $\frac{16}{31}$, respectively.