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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.