Brun-Titchmarsh theorem

The Brun–Titchmarsh theorem in analytic number theory is an upper bound on the distribution on primes in an arithmetic progression. It states that, if $$\pi(x;a,q)$$ counts the number of primes p congruent to a modulo q with p ≤ x, then


 * $$\pi(x;a,q) \le 2x / \phi(q)\log(x/q)$$

for all $$q < x$$. The result is proved by sieve methods. By contrast, Dirichlet's theorem on arithmetic progressions gives an asymptotic result, which may be expressed in the form


 * $$\pi(x;a,q) = \frac{x}{\phi(q)\log(x)} \left({1 + O\left(\frac{1}{\log x}\right)}\right)$$

but this can only be proved to hold for the more restricted range $$q < (\log x)^c$$ for constant c: this is the Siegel-Walfisz theorem.

The result is named for Viggo Brun and Edward Charles Titchmarsh.