# Brun-Titchmarsh theorem

From Citizendium

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.

## References

- Michiel Hazewinkel (2002).
*Encyclopaedia of Mathematics: Supplement 3*, 159. ISBN 0792347099. - Hugh L. Montgomery; Robert C. Vaughan (1973). "The large sieve".
*Mathematika***20**: 119-134.