Brun-Titchmarsh theorem

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For coprime integers and , let denote the number of primes not exceeding that are congruent to modulo . Using analytic methods of the theory of -functions [a8], one can show that the asymptotic formula

holds uniformly for , where is an arbitrary positive constant. It is desirable to extend the validity range for of this formula, in view of its applications to classical problems. The generalized Riemann hypothesis (cf. Riemann hypotheses) is not capable of providing any information for .

In contrast, a simple application of a sieve method [a8] leads to an upper bound which gives the correct order of magnitude of for all , where is an arbitrary positive constant. Because of its uniformity in , an inequality of this type turns out to be very useful [a3], [a5], [a8]; it is known as the Brun–Titchmarsh theorem. By a sophisticated argument, [a6], one finds that

for all . The constant possesses a significant meaning in the context of sieve methods [a2], [a7]. By adapting the Brun–Titchmarsh theorem [a1], [a4], if necessary, it is possible to sharpen the above bound in various ranges for .


[a1] E. Fouvry, "Théorème de Brun–Titchmarsh: application au théorème de Fermat" Invent. Math. , 79 (1985) pp. 383–407
[a2] H. Halberstam, H.E. Richert, "Sieve methods" , Acad. Press (1974)
[a3] C. Hooley, "Applications of sieve methods to the theory of numbers" , Cambridge Univ. Press (1976)
[a4] H. Iwaniec, "On the Brun–Titchmarsh theorem" J. Math. Soc. Japan , 34 (1982) pp. 95–123
[a5] Yu.V. Linnik, "Dispersion method in binary additive problems" , Nauka (1961) (In Russian)
[a6] H.L. Montgomery, R.C. Vaughan, "The large sieve" Mathematika , 20 (1973) pp. 119–134
[a7] Y. Motohashi, "Sieve methods and prime number theory" , Tata Institute and Springer (1983)
[a8] K. Prachar, "Primzahlverteilung" , Springer (1957)
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