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 counts the number of primes p congruent to a modulo q with p ≤ x, then
for all . 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
but this can only be proved to hold for the more restricted range for constant c: this is the Siegel-Walfisz theorem.
The result is named for Viggo Brun and Edward Charles Titchmarsh.