Let count the number of primes p congruent to a modulo q with p ≤ x. Then
for all q < x.
Historyedit
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 .
Improvementsedit
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.
brun, titchmarsh, theorem, analytic, number, theory, named, after, viggo, brun, edward, charles, titchmarsh, upper, bound, distribution, prime, numbers, arithmetic, progression, contents, statement, history, improvements, comparison, with, dirichlet, 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 Contents 1 Statement 2 History 3 Improvements 4 Comparison with Dirichlet s theorem 5 ReferencesStatement editLet p x q a displaystyle pi x q a nbsp count the number of primes p congruent to a modulo q with p x Then p x q a 2xf q log x q displaystyle pi x q a leq 2x over varphi q log x q nbsp for all q lt x History editThe 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 1 o 1 displaystyle 1 o 1 nbsp Improvements editIf q is relatively small e g q x9 20 displaystyle q leq x 9 20 nbsp then there exists a better bound p x q a 2 o 1 xf q log x q3 8 displaystyle pi x q a leq 2 o 1 x over varphi q log x q 3 8 nbsp 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 editBy contrast Dirichlet s theorem on arithmetic progressions gives an asymptotic result which may be expressed in the form p x q a xf q log x 1 O 1log x displaystyle pi x q a frac x varphi q log x left 1 O left frac 1 log x right right nbsp but this can only be proved to hold for the more restricted range q lt log x c for constant c this is the Siegel Walfisz theorem References editMotohashi 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 Retrieved from https en wikipedia org w index php title Brun Titchmarsh theorem amp oldid 1081994221, wikipedia, wiki, book, books, library,
