On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev

doi:10.5802/jtnb.186

Angel Kumchev

Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 11-23.

Résumé
Abstract

Dans cet article, nous considérons la formule asymptotique pour le nombre de représentations d’un entier impair $N$ sous la forme $p_{1} + p_{2} + p_{3} = N$ , où les $p_{i}$ sont des nombres premiers du type $p_{i} = [n^{1 / γ_{i}}]$ ; nous utilisons la méthode de van der Corput en dimension deux et nous étendons le domaine de validité de la formule asymptotique en affaiblissant les hypothèses sur les $γ_{i}$ . Dans le cas le plus intéressant $γ_{1} = γ_{2} = γ_{3} = γ$ , notre résultat entraîne que tout entier impair assez grand s’écrit comme la somme de trois nombres premiers de Piatetski-Shapiro du type $γ$ pour $50 / 53 < γ < 1$ .

In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_{1} + p_{2} + p_{3} = N$ where $N$ is an odd integer and the unknowns $p_{i}$ are prime numbers of the form $p_{i} = [n^{1 / γ_{i}}]$ . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case $γ_{1} = γ_{2} = γ_{3} = γ$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $γ$ for $50 / 53$ < $γ$ < $1$ .

MR 1469658 | Zbl 0890.11029

DOI : https://doi.org/10.5802/jtnb.186
Mots clés : Piatetski-Shapiro primes, Goldbach problem, exponential sums

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     author = {Kumchev, Angel},
     title = {On the Piatetski-Shapiro-Vinogradov theorem},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {11--23},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {1},
     year = {1997},
     doi = {10.5802/jtnb.186},
     zbl = {0890.11029},
     mrnumber = {1469658},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.186/}
}

Angel Kumchev. On the Piatetski-Shapiro-Vinogradov theorem. Journal de Théorie des Nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 11-23. doi : 10.5802/jtnb.186. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.186/

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