Brun–Titchmarsh theorem

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.

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:

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

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.

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.

• 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], "Brun-Titchmarsh theorem", Encyclopedia of Mathematics, EMS Press
• Montgomery, H.L.; Vaughan, R.C. (1973), "The large sieve", Mathematika, 20 (2): 119–134, doi:10.1112/s0025579300004708, hdl:2027.42/152543.