A logarithmic improvement in the Bombieri–Vinogradov theorem
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 635-651.

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 qx 1/2-ε .

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 qx 1/2-ε .

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Published online:
DOI: 10.5802/jtnb.1098
Classification: 11N13, 11N37, 11N60
Keywords: primes in arithmetic progressions, large sieve
Alisa Sedunova 1

1 Saint Petersburg State University 14th Line 29B, Vasilyevsky Island St.Petersburg 199178, Russia
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Alisa Sedunova. A logarithmic improvement in the Bombieri–Vinogradov theorem. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 635-651. doi : 10.5802/jtnb.1098. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1098/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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