On the Brun-Titchmarsh theorem

James Maynard. "On the Brun-Titchmarsh theorem." Acta Arithmetica 157.3 (2013): 249-296. <http://eudml.org/doc/279293>.

@article{JamesMaynard2013,
abstract = {The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size $x/(q^\{1/2\}ϕ(q))$ when logx/logq ≥ 8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem.},
author = {James Maynard},
journal = {Acta Arithmetica},
keywords = {primes; arithmetic progression; Linnik's constant; Brun-Titchmarsh theorem},
language = {eng},
number = {3},
pages = {249-296},
title = {On the Brun-Titchmarsh theorem},
url = {http://eudml.org/doc/279293},
volume = {157},
year = {2013},
}

TY - JOUR
AU - James Maynard
TI - On the Brun-Titchmarsh theorem
JO - Acta Arithmetica
PY - 2013
VL - 157
IS - 3
SP - 249
EP - 296
AB - The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size $x/(q^{1/2}ϕ(q))$ when logx/logq ≥ 8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem.
LA - eng
KW - primes; arithmetic progression; Linnik's constant; Brun-Titchmarsh theorem
UR - http://eudml.org/doc/279293
ER -