Dans cet article, nous considérons la formule asymptotique pour le nombre de représentations d’un entier impair sous la forme , où les sont des nombres premiers du type ; 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 . Dans le cas le plus intéressant , notre résultat entraîne que tout entier impair assez grand s’écrit comme la somme de trois nombres premiers de Piatetski-Shapiro du type pour .
In this paper we consider the asymptotic formula for the number of the solutions of the equation where is an odd integer and the unknowns are prime numbers of the form . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case our theorem implies that every sufficiently large odd integer may be written as the sum of three Piatetski-Shapiro primes of type for < < .
Mots clés : Piatetski-Shapiro primes, Goldbach problem, exponential sums
@article{JTNB_1997__9_1_11_0, 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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