Brun–Titchmarsh theorem

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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], "b/b110970", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Montgomery, H.L.; Vaughan, R.C. (1973), "The large sieve", Mathematika, 20 (2): 119–134, doi:10.1112/s0025579300004708.