A logarithmic improvement in the Bombieri–Vinogradov theorem
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 qx 1/2-ε .

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 .

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DOI : https://doi.org/10.5802/jtnb.1098
Classification : 11N13,  11N37,  11N60
Mots clés : primes in arithmetic progressions, large sieve
@article{JTNB_2019__31_3_635_0,
     author = {Alisa Sedunova},
     title = {A logarithmic improvement in the {Bombieri{\textendash}Vinogradov} theorem},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {635--651},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1098},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1098/}
}
Alisa Sedunova. A logarithmic improvement in the Bombieri–Vinogradov theorem. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 635-651. doi : 10.5802/jtnb.1098. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1098/

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