Primes in arithmetic progressions 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 $(a,q)=1$. Let $\chi$ be a primitive character modulo $D$, and let $\nu >0$ be small. We prove that if $L(s,\chi )$ has a Siegel zero at $s=\beta =1-\frac{1}{\eta \log D}$ with $\eta >{\eta }_0(\nu )$ for some large ${\eta }_0(\nu )$, there exists a range of $x$ for which the asymptotic $$\psi (x,q,a)=\frac{\psi (x)}{\varphi (q)}\left[1+O({\epsilon }_{{\eta }_0})-\chi \left(\frac{aD}{(q,D)}\right)\right]$$ holds for $q<x^{\frac{30}{59}-\nu }$. 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 $Q<x^{\frac{16}{31}-\nu }$. This improves on a 2003 result of Friedlander and Iwaniec that requires $q<x^{\frac{233}{462}}$ and builds on recent work of Sachpazis.

arXiv commentThis is a reworking of my previous paper (https://arxiv.org/abs/2311.12470) on Siegel zeroes and primes in arithmetic progressions in light of Sachpazis' new work (https://arxiv.org/abs/2511.16452)