A logarithmic improvement in the Bombieri–Vinogradov theorem

Sedunova, Alisa

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 635-651.

Le théorème de Bombieri–Vinogradov est l’un des outils fondamentaux de la théorie analytique de nombres ; ses applications sont nombreuses et ne se limitent pas à ce seul domaine. Dans cet article, nous améliorons la meilleure version actuellement connue du théorème, établie par Dress–Iwaniec–Tenenbaum [4], en remplaçant ${(logx)}^2$ par ${(logx)}^{5/2}$. Nous utilisons une version pondérée de l’identité de Vaughan, ce qui nous permet de faire une troncature lisse, et une estimation de Barban–Vehov [2], généralisée par Graham [6], qui est liée au crible de Selberg. Nous donnons des versions effective et non effective du résultat. En excluant les petits modules, cela nous permet de déduire un théorème de Bombieri–Vinogradov complètement effective pour $q\le x^{1/2-\epsilon }$.

The Bombieri–Vinogradov theorem is one of the standard, basic tools of an analytic number theorist; its applications are many, and not limited to the field. In this paper, we improve on the strongest version to date by Dress–Iwaniec–Tenenbaum [4], getting ${(logx)}^2$ instead of ${(logx)}^{5/2}$. We use a weighted form of Vaughan’s identity, allowing a smooth truncation inside the procedure, and an estimate of Barban–Vehov [2] (later generalized by Graham [6]), which is related to Selberg’s sieve. We give effective and non-effective versions of the result. Using that and excluding the small moduli one can derive the fully effective Bombieri–Vinogradov theorem for $q\le x^{1/2-\epsilon }$.

Classification : 11N13, 11N37, 11N60
Mots-clés : primes in arithmetic progressions, large sieve

[1] Akbary, Amir; Hambrook, Kyle A variant of the Bombieri–Vinogradov theorem with explicit constants and applications, Math. Comput., Volume 84 (2015) no. 294, pp. 1901-1932 | DOI | MR | Zbl

[2] Barban, Mark B.; Vekhov, P. P. An extremal problem, Tr. Mosk. Mat. O.-va, Volume 18 (1968), pp. 83-90 | MR | Zbl

[3] Cojocaru, Alina Carmen; Murty, M. Ram An introduction to sieve methods and their applications, London Mathematical Society Student Texts, 66, Cambridge University Press, 2006 | MR | Zbl

[4] Dress, François; Iwaniec, Henryk; Tenenbaum, Gérald Sur une somme liée à la fonction de Möbius, J. Reine Angew. Math., Volume 340 (1983), pp. 53-58 | Zbl

[5] Frolenkov, Dmitriĭ A; Soundararajan, Kannan A generalization of the Pólya–Vinogradov inequality, Ramanujan J., Volume 31 (2013) no. 3, pp. 271-279 | DOI | Zbl

[6] Graham, Sidney An asymptotic estimate related to Selberg’s sieve, J. Number Theory, Volume 10 (1978), pp. 83-94 | DOI | MR | Zbl

[7] Helfgott, Harald The ternary Goldbach problem (2015) (https://arxiv.org/abs/1501.05438, submitted) | Zbl

[8] Landau, Edmund Über Ideale und Primideale in Idealklassen, Math. Z., Volume 2 (1918) no. 1-2, pp. 52-154 | DOI | Zbl

[9] Lenstra, Hendrik W.; Pomerance, Carl Primality testing with Gaussian periods, J. Eur. Math. Soc., Volume 21 (2019) no. 4, pp. 1229-1269 | DOI | MR | Zbl

[10] Liu, H.-Q. An effective Bombieri–Vinogradov theorem and its applications, Acta Math. Hung. (2017), pp. 230-235 | DOI | MR | Zbl

[11] Montgomery, Hugh L. The analytic principle of the large sieve, Bull. Am. Math. Soc., Volume 84 (1978) no. 4, pp. 547-567 | DOI | MR | Zbl

[12] Page, A. On the number of primes in an arithmetic progression, Proc. Lond. Math. Soc. (1935), pp. 116-141 | DOI | MR | Zbl

[13] Sedunova, Alisa A partial Bombieri–Vinogradov theorem with explicit constants, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 2018 (2018), pp. 101-110 | DOI | MR | Zbl

[14] Timofeev, Nikolaĭ M The Vinogradov–Bombieri theorem, Mat. Zametki, Volume 38 (1985) no. 6, pp. 801-809 | MR | Zbl

[15] Vinogradov, Ivan M. The method of trigonometrical sums in the theory of numbers, Dover Publications, 1954, x+180 pages (reprint of the 1954 translation) | Zbl