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
[edit]Let count the number of primes p congruent to a modulo q with p ≤ x. Then
for all q < x.
History
[edit]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 .
Improvements
[edit]If q is relatively small, e.g., , then there exists a better bound:
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
[edit]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 q < (log x)c for constant c: this is the Siegel–Walfisz theorem.
References
[edit]- 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.