## On the Brun–Titchmarsh theorem

### Volume 157 / 2013

Acta Arithmetica 157 (2013), 249-296
MSC: 11N13, 11N05, 11M06, 11M20.
DOI: 10.4064/aa157-3-3

#### 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 $(C+o(1))x/(\phi(q)\log{x})$ 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.