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.

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

for all q < x.

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

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

...
Wikipedia