In
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
, the Brun–Titchmarsh theorem, named after
Viggo Brun and
Edward Charles Titchmarsh
Edward Charles "Ted" Titchmarsh (June 1, 1899 – January 18, 1963) was a leading British mathematician.
Education
Titchmarsh was educated at King Edward VII School (Sheffield) and Balliol College, Oxford, where he began his studies in October 1 ...
, is an
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of .
Dually, a lower bound or minorant of is defined to be an element of that is less ...
on the distribution of
prime numbers in arithmetic progression.
Statement
Let
count the number of primes ''p'' congruent to ''a'' modulo ''q'' with ''p'' ≤ ''x''. Then
:
for all ''q'' < ''x''.
History
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
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
In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s.
Description
In ...
, 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
By contrast,
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is al ...
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 In analytic number theory, the Siegel–Walfisz theorem was obtained by Arnold Walfisz as an application of a theorem by Carl Ludwig Siegel to primes in arithmetic progressions. It is a refinement both of the prime number theorem and of Dirichlet ...
.
References
*
*
*
* .
{{DEFAULTSORT:Brun-Titchmarsh theorem
Theorems in analytic number theory
Theorems about prime numbers