Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes

TL;DR

This paper investigates the distribution of primes in arithmetic progressions for large moduli under the assumption of an exceptional zero of a Dirichlet L-function, extending previous results to larger ranges of moduli.

Contribution

It establishes prime distribution asymptotics for large moduli under weaker assumptions on the exceptional zero, improving the range of moduli where results hold.

Findings

Asymptotic formula for primes in arithmetic progressions with large moduli

Results hold for almost all residue classes and moduli within a larger range

Extension of previous bounds on the size of moduli under Siegel zero assumptions

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.