Brun-Titchmarsh theorem

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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 ${\displaystyle \pi (x;a,q)}$ counts the number of primes p congruent to a modulo q with px, then

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

for all ${\displaystyle q. 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

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

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

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