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 when logx/logq ≥ 8 and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa157-3-3,
author = {James Maynard},
title = {On the Brun-Titchmarsh theorem},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {249-296},
zbl = {1321.11099},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa157-3-3}
}
James Maynard. On the Brun-Titchmarsh theorem. Acta Arithmetica, Tome 161 (2013) pp. 249-296. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa157-3-3/