# Brun–Titchmarsh theorem

Brun–Titchmarsh 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 References Statement 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. 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 {displaystyle 1+o(1)} . Improvements If q is relatively small, e.g., {displaystyle qleq 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. Comparison with Dirichlet's theorem 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+Oleft({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. References 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. Categories: Theorems in analytic number theoryTheorems about prime numbers

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