On the Piatetski-Shapiro-Vinogradov theorem

Kumchev, Angel

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

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 $p_i=\left[n^{1/{\gamma }_i}\right]$ ; 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 ${\gamma }_i$. Dans le cas le plus intéressant ${\gamma }_1={\gamma }_2={\gamma }_3=\gamma$, notre résultat entraîne que tout entier impair assez grand s’écrit comme la somme de trois nombres premiers de Piatetski-Shapiro du type $\gamma$ pour $50/53<\gamma <1$.

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 $p_i=\left[n^{1/{\gamma }_i}\right]$. We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case ${\gamma }_1={\gamma }_2={\gamma }_3=\gamma$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $\gamma$ for $50/53$ < $\gamma$ < $1$.

Mots clés : Piatetski-Shapiro primes, Goldbach problem, exponential sums

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