Primes in arithmetic progressions and Siegel zeroes
Abstract
Let χ be a Dirichlet character mod D with L(s,χ) its associated L-function, and let ψ(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 χ be a primitive character modulo D, and let ν>0 be small. We prove that if L(s,χ) has a Siegel zero at s=β=1−1/(ηlog D) with η>η0(ν) for some large η0(ν), there exists a range of x for which the asymptotic ψ(x,q,a)=ψ(x)/ϕ(q)[1+O(εη0)−χ(aD/(q,D))] holds for q<x30/59−ν. 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∼ Q on the range Q<x16/31−ν. This improves on a 2003 result of Friedlander and Iwaniec that requires q<x233/462 and builds on recent work of Sachpazis.
Related
- Cyclicity of Drinfeld modules modulo primes in arithmetic progressions
- Diophantine approximation by primes and Landau--Siegel zeros
- Triple divisor type functions in arithmetic progressions to prime power moduli
- Hitting Arithmetic Progressions at the Square-Root Scale
- Arithmetic Siegel--Weil formula on the product of modular curves
BibTeX
@misc{wright2026primes,
title = {{Primes in arithmetic progressions and Siegel zeroes}},
author = {Thomas Wright},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35950},
eprint = {2609.35950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35950},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Primes in arithmetic progressions and Siegel zeroes
AU - Thomas Wright
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35950
UR - https://arxiv.org/abs/2609.35950
AB - Let χ be a Dirichlet character mod D with L(s,χ) its associated L-function, and let ψ(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 χ be a primitive character modulo D, and let ν>0 be small. We prove that if L(s,χ) has a Siegel zero at s=β=1−1/(ηlog D) with η>η_0(ν) for some large η_0(ν), there exists a range of x for which the asymptotic ψ(x,q,a)=ψ(x)/ϕ(q)[1+O(ε_η_0)−χ(aD/(q,D))] holds for q<x^30/59−ν. 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∼ Q on the range Q<x^16/31−ν. This improves on a 2003 result of Friedlander and Iwaniec that requires q<x^233/462 and builds on recent work of Sachpazis.
ER -