Brun–Titchmarsh theorem
# Brun–Titchmarsh theorem

In analytic number theory, the Brun–Titchmarsh theorem, named after Viggo Brun and Edward Charles Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression.

## Statement

Let $\pi (x;q,a)$ count the number of primes p congruent to a modulo q with p ≤ x. Then

    $\pi (x;q,a)\leq {2x \over \varphi (q)\log(x/q)}$

for all q < x.

## History

The result was proven by sieve methods by Montgomery and Vaughan; an earlier result of Brun and Titchmarsh obtained a weaker version of this inequality with an additional multiplicative factor of $1+o(1)$.

## Improvements

If q is relatively small, e.g., $q\leq x^{9/20}$, then there exists a better bound:

    $\pi (x;q,a)\leq {(2+o(1))x \over \varphi (q)\log(x/q^{3/8})}$

This is due to Y. Motohashi (1973). He used a bilinear structure in the error term in the Selberg sieve, discovered by himself. Later this idea of exploiting structures in sieving errors developed into a major method in Analytic Number Theory, due to H. Iwaniec's extension to combinatorial sieve.

## Comparison with Dirichlet's theorem

By contrast, Dirichlet's theorem on arithmetic progressions gives an asymptotic result, which may be expressed in the form

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

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.

