On the Brun-Titchmarsh Theorem

Details On the Brun-Titchmarsh Theorem On the Brun-Titchmarsh Theorem

2012-01-09

arxiv.org/abs/1201.1777v1

Mathematics

Number Theory

47 Pages

Scientific paper

The Brun-Titchmarsh theorem shows that the number of primes $\le x$ which are congruent to $a\pmod{q}$ is $\le (C+o(1))x/(\phi(q)\log{x})$ 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 $C>2$. We show that one can take C=2 provided that $\log{x}/\log{q}\ge 8$. Without excluding the possibility of an exceptional Siegel zero, we cannot have $C<2$ and so this result is best-possible in this sense. We obtain this result using analytic methods developed in the study of Linnik's constant. In particular, we obtain explicit bounds on the number of zeroes of Dirichlet $L$-functions with real part close to 1 and imaginary part of size O(1).

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