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 instead of . 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 .
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 par . 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 .
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1098
Keywords: 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}, zbl = {06984511}, mrnumber = {4102618}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1098/} }
TY - JOUR AU - Alisa Sedunova TI - A logarithmic improvement in the Bombieri–Vinogradov theorem JO - Journal de théorie des nombres de Bordeaux PY - 2019 SP - 635 EP - 651 VL - 31 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1098/ DO - 10.5802/jtnb.1098 LA - en ID - JTNB_2019__31_3_635_0 ER -
%0 Journal Article %A Alisa Sedunova %T A logarithmic improvement in the Bombieri–Vinogradov theorem %J Journal de théorie des nombres de Bordeaux %D 2019 %P 635-651 %V 31 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1098/ %R 10.5802/jtnb.1098 %G en %F JTNB_2019__31_3_635_0
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/
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