On the Brun–Titchmarsh theorem

James Maynard¹

¹ Mathematical Institute 24-29 St Giles' Oxford, OX1 3LB, England

Acta Arithmetica, Tome 157 (2013) no. 3, pp. 249-296.

Résumé

The Brun–Titchmarsh theorem shows that the number of primes which are less than $x$ and congruent to $a$ modulo $q$ is less than $(C+o(1))x/(\phi(q)\log{x})$ for some value $C$ depending on $\log{x}/\!\log{q}$. Different authors have provided different estimates for $C$ in different ranges for $\log{x}/\!\log{q}$, all of which give $C>2$ when $\log{x}/\log{q}$ is bounded. We show that one can take $C=2$ provided that $\log{x}/\log{q}\ge8$ and $q$ is sufficiently large. Moreover, we also produce a lower bound of size $x/(q^{1/2}\phi(q))$ when $\log{x}/\log{q}\ge 8$ and is bounded. Both of these bounds are essentially best-possible without any improvement on the Siegel zero problem.

Détail

DOI : 10.4064/aa157-3-3

Keywords: brun titchmarsh theorem shows number primes which congruent modulo phi log value depending log log different authors have provided different estimates different ranges log log which log log bounded provided log log sufficiently large moreover produce lower bound size phi log log bounded these bounds essentially best possible without improvement siegel zero problem