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

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

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 ${\displaystyle 1+o(1)}$.

## Improvements

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

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

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

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

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.

## References

• Motohashi, Yoichi (1983), Sieve Methods and Prime Number Theory, Tata IFR and Springer-Verlag, ISBN 3-540-12281-8
• Hooley, Christopher (1976), Applications of sieve methods to the theory of numbers, Cambridge University Press, p. 10, ISBN 0-521-20915-3
• Mikawa, H. (2001) [1994], "b/b110970", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Montgomery, H.L.; Vaughan, R.C. (1973), "The large sieve", Mathematika, 20 (2): 119–134, doi:10.1112/s0025579300004708.