Primes in arithmetic progressions under the presence of Landau-Siegel zeroes
Stelios Sachpazis
TL;DR
This work analyzes primes in arithmetic progressions under the presence of Landau–Siegel zeroes, establishing a conditional asymptotic for ψ(x;q,a) in a broader range of moduli than GRH allows. The authors replace Λ with a Möbius-like convolution λ′=χ∗log under exceptional characters and reduce the problem to sifted sums over z-rough numbers using a preliminary sieve, additive characters, and Heath–Brown type techniques. Central contributions include a robust asymptotic for sums of λ over progressions with q up to x^{2/3−ε} (improving over the square-root barrier) and a carefully controlled error term that depends on the Landau–Siegel zero quality η and the auxiliary parameter V. The results demonstrate how Landau–Siegel zeros influence primes in progressions and provide a flexible framework that can be improved further by refining sieve weights and exponential-sum estimates.
Abstract
Let $x\geqslant 2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $ψ(x;q,a):=\sum_{n\leqslant x,n\equiv a(\!\!\!\mod{\!\!q})}Λ(n)$, where $Λ$ is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of exceptional characters corresponding to "extreme" Landau-Siegel zeroes and established a meaningful asymptotic formula for $ψ(x;q,a)$ beyond the square-root barrier of the Generalized Riemann Hypothesis. In particular, their asymptotic yields non-trivial information for moduli $q\leqslant x^{1/2+1/231}$. In this paper, we considerably relax the extremity of the Landau-Siegel zero required in the work of Friedlander and Iwaniec and obtain a conditional asymptotic formula for $ψ(x;q,a)$ in a slightly wider range of $q$.