Teorema di Brun-Titchmarsh

Teorema di Brun-Titchmarsh In teoria analitica dei numeri, il teorema di Brun-Titchmarsh, prende il nome da Viggo Brun e Edward Charles Titchmarsh, è un limite superiore della distribuzione dei numeri primi nella progressione aritmetica.
Contenuti 1 Dichiarazione 2 Storia 3 Miglioramenti 4 Confronto con il teorema di Dirichlet 5 Riferimenti Dichiarazione Let {stile di visualizzazione pi (X;q,un)} contare il numero di primi p congruenti ad a modulo q con p ≤ x. Quindi {stile di visualizzazione pi (X;q,un)leq {2x su varphi (q)tronco d'albero(x/q)}} per tutte 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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