Geodesic
The Bombieri-Vinogradov theorem for nilsequences
Discrete analysis (2021)
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We establish results of Bombieri-Vinogradov type for the von Mangoldt function $Λ(n)$ twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase $e(P(n))$; the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes $p$ obeying a "nil-Bohr set" condition, such as $\|αp^k\|\varepsilon$, exhibit bounded gaps. Secondly, we show that the Chen primes are well-distributed in nil-Bohr sets, generalizing a result of Matomäki. Thirdly, we generalize the Green-Tao result on linear equations in the primes to primes belonging to an arithmetic progression to large modulus $q\leq x^θ$, for almost all $q$.
Publié le : 
@article{DAS_2021_a5,
     author = {Xuancheng Shao and Joni Ter\"av\"ainen},
     title = {The {Bombieri-Vinogradov} theorem for nilsequences},
     journal = {Discrete analysis},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2021_a5/}
}
TY  - JOUR
AU  - Xuancheng Shao
AU  - Joni Teräväinen
TI  - The Bombieri-Vinogradov theorem for nilsequences
JO  - Discrete analysis
PY  - 2021
UR  - http://geodesic.mathdoc.fr/item/DAS_2021_a5/
LA  - en
ID  - DAS_2021_a5
ER  - 
%0 Journal Article
%A Xuancheng Shao
%A Joni Teräväinen
%T The Bombieri-Vinogradov theorem for nilsequences
%J Discrete analysis
%D 2021
%U http://geodesic.mathdoc.fr/item/DAS_2021_a5/
%G en
%F DAS_2021_a5
Xuancheng Shao; Joni Teräväinen. The Bombieri-Vinogradov theorem for nilsequences. Discrete analysis (2021). http://geodesic.mathdoc.fr/item/DAS_2021_a5/