On the Piatetski-Shapiro-Vinogradov theorem
Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 11-23.

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.

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.

Keywords: Piatetski-Shapiro primes, Goldbach problem, exponential sums
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Kumchev, Angel. On the Piatetski-Shapiro-Vinogradov theorem. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 11-23. http://archive.numdam.org/item/JTNB_1997__9_1_11_0/

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